###
*******For a specific day, press "Ctrl+f " (for a PC) and type in the day you would like.********

###
Other Number/Calendar Blogs

###
October 1st (274th Day of the Non-Leap Year)

Oct 1st, 10/1.

101 = 10^(1+0+1) + 1

101 = (10-1)*(10+1) + (1+0+1)

101 = (10+1)*(11) - 10*(1+0+1)

101 = 110 - (10-1)

101 = 10*(10) + 1

101, 101+2, 101+6, 101+8, and 101+12 are all prime. (What's the next number with the same property?)

101 is the second palindromic prime (excluding one digit primes).

101 - (1+0+1) = 99 is a palindrome.

101 + (1+0+1) = 103 is a prime.

(What other numbers have these properties?)

101, 101^2, 101^3, and 101^4 are all palindromes. (Is there another number with this property?)

101*11 = 101*(10+1) = 1111 (repdigit)

101 is the first prime and the first palindrome with a 0 in it. (What is another number with both these properties?)

3^101 - 2^101 is prime. (What is the next number following this pattern?)

101 is the first prime with only 0s and 1s. (What is the next prime?)

101^2 + 101 + 1 is prime.

101^2 - 101 - 1 is prime. (What's the next number satisfying both these properties?)

101 is 5 in base 2 (5 is prime and 2 is prime). (Do other numbers share this property?)

101 is the smallest 3-digit prime. (What is the smallest 4-digit prime? 5-digit? 6?)

101 = 79 + 22 (22nd prime plus 22).

10^101 + 3 is prime. (What's the next number with this property?)

101^1 has a digit sum of (1+0+1)^1.

101^2 has a digit sum of (1+0+1)^2.

101^3 has a digit sum of (1+0+1)^3.

101^4 has a digit sum of (1+0+1)^4.

(Do other numbers have this same property?)

There are 101 primes less than 550.

101 = 10^2 - 1^2 (Note: 10 and 1 make up 101). (Are there other numbers with this property?)

101 = 13 + 17 + 19 + 23 + 29 (5 consecutive primes).

101 is the only known prime to exist such that its digit alternate between 1 and 0.

101 is the largest known prime of the form 10^n + 1.

Room 101 is the torture chamber in George Orwell's book 1984.

101 is the generic term used for an intro course in college (Math 101, Art 101, Psych 101, etc.)

274th Day of the Year.

274 = 27 + 74 + 24 + 2*7*4 + (2+7+4) + 74 + (2+4)

274 = 47 + 72 + 42 + 2*7*4 + (27+4) + (24-7) - (2-7-4)

274 = 27*4 + 2*74 + (2*7+4)

274 = 47*2 + 4*72 - 27*4

274 = 247 + 27

274 = (24-7) + (27-4) + (72-4) + (74-2) + (47-2) + (42-7) + (7*4)/2

274^2 = 75076 (ends in consecutive numbers). (Do other numbers share this property?)

274^64 + 1 is prime. (What's the next number that shares this property?)

There are 274 primes less than 42^2.

There are 274 primes between 257^2 and 263^2 (the 55th prime squared and the 56th prime squared).

274 = 13*19 (product of two primes)

274 = 107 + 167 (sum of two primes) (What's the next number with the same properties?)

2740123456789 is prime. (What other numbers have this property?)

274 = 2*137

2 + (1+3+7) = 2+7+4. (What is the next number with this property?)

###
October 2nd (275th Day of the Non-Leap Year)

Oct 2nd, 10/2.

102 = (10+2)*(10-2) + 2*(1+0+2)

102 = 10 + 2 + 12 + 20 + 1 + 21 + (10/2 + 1)^2

102 = (20+1)*(20-1) - 201 - (10+2)*(10-2)

102 = 12 + 34 + 56

102 - (1+0+2) = 1 + 2 + 3 + 4 + 5 + 6 + 78

102*201 = 20502 (a palindrome). (What are other numbers with the same property?)

102 is the smallest 3-digit number to have 3 prime divisors.

102^2 + 103^2 is prime.

Numbers 102,103,104,105, when squared, have a DigitSum that is also a square. (100 and 101 also share this property. Is there another run of 6 numbers that share this property?)

There are 102 primes less than 560.

The sum of the first 102 primes is prime. (Does this work for other numbers?)

102^64 + 1 is prime. (What are other numbers that share this property?)

102!!!!!! + 1 is prime (six factorials). (What's another number with this property?)

There are 102 primes between 14000 and 15000.

There are 102 primes between 42000 and 43000.

102 = 19 + 23 + 29 + 31 (four consecutive primes)

The Empire State Building in New York City has 102 floors.

275th Day of the Year.

275 = 27 + 75 + 25 + 2*75 - (7-5)

275 = 57 + 72 + 52 + 2*7*5 + (27-5) + 2

275 = 27*5 + 2*75 + (2-7-5)

275 = (27-5) + (25-7) + (52-7) + (57-2) + (75-2) + (72-5) - 5

275 = 25*7 + (2+7+5) + 2*7*5 + (2*7-5) + 7

For any positive integer N, 275^N contains a 2, 7, and 5. (Is there another number like this?)

275*572 = 157300 (two 0s appear from numbers with nonzero digits) (What other numbers have this property?)

275 = 5*5*11

275*(5+5+11) = 5775 (palindrome) (Are there other numbers with this property?)

(Note: 5*5*11 can be 5*11*5, a palindromic expression).

The digits of 275 are all prime.

The OEIS has 2325 results found for the number 275.

The digits of 2325 are also all prime. (Do other numbers have this property?)

275 = 2^5 + 3^5 (Note: all prime numbers are used.) (Are there other numbers with these properties?)

275^16 + 276^16 is prime. (What's another number with the same property?)

275 = 55*5 (only uses 5 to express itself).

276^11 - 275^11 is a prime. (What's the next number with the same property?)

275 is a composite number with prime digits. (What's the next number with this property?)

275^3 has an equal number of odd and even digits. (What's another number with this property?)

###
October 3rd (276th Day of the Non-Leap Year)

Oct 3rd, 10/3.

103 = 10 + 13 + 3 + 30 + 31 + 1 + (10+3) - (1-0-3)

103 = (10+3)*(10-3) + (10+3) - 1

103 = (10-3)*(10-3) - (1-0-3)*(10+3) + 30 + (1-0-3)

103 = 130 - 3^(3+0*1)

103 = 310 - 3*10 - (31+(30-(3-1)))*3

103 = 301 - 30*1 - (3-0-1)*(10^(3-1)-(3-1)^(3+1))

103 = 31*3*1 + 30/3

103 = 13*1*3 + (3+1)^3

103 = 31*(3+1) - 3*(10-3)

103 = 13*(1+3) + 3*(30-13)

103 = (10+3)^(3-1) - 3*((3+1)!-(3-1))

103 = (1+0+3)*(30-3-1) - (3-3^0-1)

103*301 = 31003 (Only 1,0,3 are used). (Are there other numbers with the same property?)

103*13 and 103*31 both have the same digits (in different orders). (Does this hold for other numbers?)

Inserting a 1 anywhere in 103 retains the prime property. (i.e. 1103, 1013, 1031 are all prime.) (What is the another number that shares this property?)

There are 103 primes between 11000 and 12000.

The world's largest tree, the General Sherman Tree, has a trunk that is roughly 103 feet in circumference.

103 + (10!)^3 is prime. (Are there other numbers with this property?)

Using a standard dartboard, 103 is the lowest prime number that cannot be scored with two darts.

103 is a strictly non-palindromic number (that is, it isn't a palindrome in bases 2 through 101). (What is the next number with the same property?)

When "and" is included, 103 is the smallest number requiring 18 letters when spelled out in English.

Moving the 3 to the front, 310 = 103*3 + 1.

276th Day of the Year.

276 = 27 + 76 + 26 + 2*76 - (2*6-7)

276 = 67 + 72 + 62 + 2*7*6 - (2+7)

276 = 27*6 + 2*76 - (2*6-7) - (27+6)

276 = 726 - 7*26 - 67*(7-6/2)

276 = 267 + 27/(6/2)

276 = 762 - 76*2 - 7*6*2 - (7+6/2)*(27-2)

276 = 627 - 62*7 + (76+7)

276 = 672 - 6*72 + (27+2+7)

276 = (2+76) + (27+6) + (67+2) + (6+72) + (2+7+6) + (2+7-6)

276 = 2*76 + 27*6 - (26-7)*2

276 = 6*72 - 67*2 - (27-(7-2))

276 = (2+76) + (27+6) + (7-2)*(27+6)

276 = (6+72) + (67+2) + (6/2)*(27+2^(6-2))

276 = (2+7+6)*(26-7) - (7*2-(7-2))

276 = (2*7+6)*2*7 - (7-6/2)

276 = (2+7*6)*6 + (7*6-6*(7-2))

276 = (2+76)*(6/2) + (27+(7-2)*(6/2))

276 = (27+6)*(2*7-6) + (2+7+6/2)

276 = (27-6)*(7+6) + (2+7-6)

276^2 = 76176 (starts/ends with 76 and 76 is in the decimal representation of 276). (Are there other numbers with this property?)

276*672 = 185472 (all different digits). (What other numbers have the same property?)

276*96 ends in a 96.

276*76 ends in a 76.

276*56 ends in a 56.

276*36 ends in a 36.

276*16 ends in a 16.

(Note: 9,7,5,3,1 are all odd digits).

(The same property holds if we used 12,32,52,72,92).

(Does this hold for other numbers?)

2*76 + 27*6 = 314 (the beginning digits of pi).

276 = 2*2*3*23 (only 2 and 3 are used). (What other numbers only use two numbers in their prime factors?)

276 = 1^5 + 2^5 + 3^5

276^4 + 1 is prime. (What are other numbers that share this property?)

There are 276 13-digit left-truncatable primes (removing the leftmost digit of a prime number remains prime).

There are 276 ways to place 3 non-attacking kings on a 4 by 4 board.

6*276 + 1, 12*276 + 1, and 18*276 + 1 are all prime. (What other numbers have the same property?)

276 is the smallest number whose aliquot sequence has not yet been determined. (What is the next number with this property?)

276 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 (12 consecutive primes).

The highest number of rounds in boxing is 276, the boxing match took 4 and a half hours.

The sum of divisors of 276 is 672 (reverse of 276). (What other numbers have this property?)

###
October 4th (277th Day of the Non-Leap Year)

Oct 4th, 10/4.

104 = 10 + 14 + 40 + 41 + (1+0+4) - (10-4)

104 = (10-4)*(10+4) + (10-4) + 14

104 - (1+0+4) = 9 + 8 + 7 + 6 + 5 + 43 + 21

104^3 and 104^4 do not contain zeroes. (Do other numbers N have this property given that N has at least one zero?)

104*401 has 1,0,4 (in order) in its decimal representation. (Are there others with the same property?)

104*(10+4) = 1456 (1 < 4 < 5 < 6) (Does the property hold for other numbers?)

104^2 + 105^2 is prime. (What are other numbers like this?)

There are 104 primes less than 570.

90*104 + 73 and 90*104 + 37 are both prime. (What other numbers have both these properties?)

104^23 + 104^21 + 104^19 + 104^17 + 104^15 + 104^13 + 104^11 + 104^9 + 104^7 + 104^5 + 104^3 + 104 + 1 is prime. (What's the next number with this property?)

There are 104 primes in [17000,18000], [19000,20000] and [21000,22000]. (Do other numbers share this property?)

The 104th and 105th primes are twin primes. (Are there other numbers like this?)

104 has 8 divisors (8 being one of them). (What's the next number with the same property?)

104 is a palindrome in base 5 (404), base 6 (252), and base 12 (88). (Are there other numbers that have this property?)

277th Day of the Year.

277 = 27 + 77 + 72 + 2*7*7 + (2+7/7)

277 = 27*7 + 2*77 + (2-77) + (2+7)

277 = (27-7) - (7+7-2) + 77 - (2+7+7)*(2-7-7)

277 = (2+7)^2 + (7+7)^2

277^2 and 277^3 both begin with a three digit palindrome. (What are others that share this property?)

277^4 = 5887339441 (has pattern ABBCDDEFF...) (Does this property hold for other numbers?)

277^2 contains 2,7,7 in reverse order of 277. (Are there others like this?)

277*(2+7+7) = 4432.

277*(2-7-7) = -3324.

(Same numbers used). (Do other numbers have this property?)

277 is prime.

3^277 - 2^277 is prime, as well. (What are other numbers with this property?)

277 is a prime number with prime digits. (What's the next number with this property?)

277 is the smallest prime containing 2 sevens.

277 has two 7s (Note the phrasing of this sentence.) (What other numbers have this property?)

277 is the largest prime factor of 1 + 2*3*5*7*11*13*17.

277 = 1^5 + 1^5 + 2^5 + 3^5 (first four Fibonacci numbers to the fifth power)

277, 277 + 4^1, and 277 + 4^2 are prime. (Do other numbers share this property?)

277 is the smallest prime factor of 1 + 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59.

277 is the 59th prime. 59 is the 17th prime. 17 is the 7th prime. (7 is prime, as well). (Are there other numbers that are like this?)

277 is the smallest prime such that 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... + 1/277 > 2.

(Note: 1/2 + 1/3 + 1/5 is already > 1).

The maximum pieces of pizza you can have after 23 straight cuts is 277.

On a large enough chessboard, there are 277 squares a knight can reach in exactly six moves.

277 appears in the fifth term of the Taylor expansion for sec(x) (277x^8/8064).

The Grand Canyon is 277 miles long.

###
October 5th (278th Day of the Non-Leap Year)

Oct 5th, 10/5.

105 = 15 + 51 + (50-1) - 10

105 = (10+5)*(10-5) + (10+5)*(10/5)

105 = (1+0+5)*(10+5) + 15

105 = 3*5*7 (first three odd primes).

1 + 105 + 105^2 is prime.

10^105 + 3 is prime.

There are 105 primes less than 24^2.

There are 105 primes in [13000,14000].

105 is palindrome in base 4 (1221), base 8 (151), and base 20 (55).

105 = 7!! (7*5*3*1)

105 is in the middle of a prime quadruplet (101, 103, 107, 109).

"105" is a card game where the player loses if he or she has over 105 points.

The positive values of 105 - 2^k are all prime (for all k that make this positive).

105/pi^4 = (1+1/2^4)*(1+1/3^4)*(1+1/5^4)*(1+1/7^4)*(1+1/11^4)*...

278th Day of the Year.

278 = 27 + 78 + 28 + 2*7*8 + (27+8) + (2*(7-8))

278 = 87 + 72 + 82 + (27+8) - (2*(7-8))

278 = 27*8 + 2*78 + (2-78) + (2+7+8) + (2+7-8)

278 = (27+8) + (2+78) + (87+2) + (8+72) - (2*7-8)

278 = (72-8) - (2+7+8)*(2-7-8) - 7

277^2 and 278^2 contain 2,7,7 and 2,7,8 respectively.

278^4 + 1 is prime.

There are 278 primes less than 1800.

278^256 + 1 is prime.

1789 - 278 is prime (278th prime minus 278).

278^2 + 278 + 1 is prime.

###
October 6th (279th Day of the Non-Leap Year)

Oct 6th, 10/6.

106 = 10 + 60 + 61 + 16 - (10+6)*(10-6) + (1+0+6) + (10+6)

106 = (10+6)*(10-6) + (1+0+6)*(10-6) + (10-6) + 10

106 = (10+6)*(1+0+6) - 6

106 - 61 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

106*106 = 11236 (1 â‰¤ 1 â‰¤ 2 â‰¤ 3 â‰¤ 6). (Are there other numbers with this property?)

106 = 83 + 23 (23rd prime plus 23).

106^4 + 1 is prime. (What's another number with this property?)

There are 106 primes less than 560.

3^106 - 10 is prime. (What's the next number with this property?)

106!!!!! + 1 is prime.

106!!!!!! + 1 is prime. (What other numbers have one or both of these properties?)

There are 106 primes in [10000,11000] and [32000,33000].

17^106 - 2 is prime.

21^106 - 2 is prime.

No other number has both these properties.

The number 9 appears 106 times in the first 1000 digits of pi.

106th prime plus 106 is a prime. (577+106 = 683)

279th Day of the Year.

279 = 27 + 79 + 29 + (2-(7+9) + (9-7))^2

279 = 97 + 72 + 92 + (2+7+9)

279 = (27+9) + (2+79) + (97+2) + (9+72) - (2+7+9)

279 = 27 - (2+7+9)*(2-7-9)

279 = 27*9 + 2*79 - (2+9)^(9-7) + 2/(7-9)

279 = 2*7*9 + 27*9 - (92-7) - (7-2)

279 = 297 - (2+9+7)

279 = 27*9 + (27+9)

279 = 229 + 50 (50th prime plus 50).

10^57 + 279 is the first prime with 58 digits.

There are 54 groups with an order equal to or less than 279.

279 is a divisor of 999,999,999,999,999 (smallest string of 9s).

279 = 8!! - 7!!

2791331 and 1331279 are both primes. (What other numbers share this property?)

Every positive number is the sum of at most 279 eighth powers.

279 = 3^2 + 3^3 + 3^5 (powers are consecutive primes).

###
October 7th (280th Day of the Non-Leap Year)

Oct 7th, 10/7.

107 = 71 + (1-0-7)*(1-0-7)

107 = 17*(1+0+7) + (1+0+7)*(1-0-7) + (10-7)*(10-7) + 10

107 = (10+7)*(10-7) + 70 - (10+7) + (10-7)

107 = 10 + 17 + 70 + 71 - 17*(10-7) - 10

107 = 2 + 3*5*7 = 2*3*5 + 7*11

107^2 has digits that are all squares. (What are other numbers with this property?)

701*107 = 75007 (Note: 701 starts with a 7 and 107 ends with a 7). (What other numbers display this property?)

107*(10+7) = 107*17 = 1819 (concatenation of two consecutive numbers. Also note that 18 and 19 come right after 17). (Are there other numbers like this?)

107^3 only uses digits 0,1,2,3,4,5. (Do other numbers have this property?)

107 and 701 are both primes. (What's the next number like this?)

There are no primes between 1070 and 1079.

10^107 + 3 is prime. (What are other numbers with this property?)

There are 107 primes less than 580.

The 107% rule in Formula One racing decides which cars are qualified.

The polynomial n^2 - n + 67374467 is never divisible by any prime less than 107 (for integers n).

2^107 reversed and 2^107 - 1 are primes.

2^107 - 1 is the largest known Mersenne prime not containing all the individual digits.

There is no integer N such that N! has 107 zeroes.

107 = 1! in base 2 + 0! in base 2 + 7! in base 2. (This is the largest prime with such property. What's the next number less than 107 like this?)

107 ("One hundred and seven") is the smallest English number needing six syllables.

107 remains prime if 2 is added to any of its digits (307, 127, 109). (Do other numbers also share this property? Instead of a 2, what about 1? 3? 4? ...)

107 + 2!, 107 + 3!, ... 107 + 9! are all prime. (Note: There are (1+0+7) = 8 numbers here.) (Are there others with this property?)

Hyperpyrexia is a severe elevation of the body temperature (usually to 107 degrees Fahrenheit).

280th Day of the Year.

280 = 28 + 80 + 20 + 82 + (80/2) + (8/2 + 0) + (8+2+0) + (8*2)

280 = 2*80 + (2+8+0)*(20-8)

280 = 208 + (2-8-0)*(8-20)

280 = (3!)^3 + (2!)^6

280^1 + 1 is prime.

280^2 + 1 is prime.

(What other numbers have both of these properties?)

280 = 2*2*2*5*7

280*(2+2+2+5+7) = 5040 = 7! (a factorial). (Are there other numbers with this property?)

10^280 + 13 is prime. (What's the next number with the same property?)

280 = (2^3)*5*7 (first four primes used in prime factorization).

280^6 + 280^5 + 280^4 + 280^3 + 280^2 + 280 + 1 is prime. (What are other numbers with this property?)

3^280 - 28 is prime (Note: 28 is a substring of 280). (What numbers also display this property?)

280!!!!! + 1 is prime. (What's the next number with this property?)

###
October 8th (281st Day of the Non-Leap Year)

Oct 8th, 10/8.

108 = 10 + 18 + 80

108 = 81 + (10-8)*(1+0+8) + (1*8) + 1

108 = 10*8 + (10+8)(10-8) - 8

108 = (10+8)*(1+0+8) - (80-1) + (10+8) - (1-0-8)

108^2 = 11664 (concatenation of three squares--1//16//64). (Are there other numbers with the same property?)

108*810 ends in 810. (Do other numbers have this property?)

108 = 2*2*3*3*3 (two 2s and three 3s).

10^108 + 19 is prime. (What's the next number with the same property?)

The sum of the first 108 primes is prime. (For what other n does this property satisfy?)

There are 108 primes in [15000,16000].

108 = (1^1)*(2^2)*(3^3)

There are 108 nonzero palindromes less than 1000.

The number 108 is very sacred in many religions. The digits 1,0,8 mean one thing, nothing, and everything (infinity).

The prehistoric monument Stonehenge is around 108 feet in diameter.

In martial arts, there are 108 pressure points in the human body.

An official MLB baseball has 108 stitches.

There are 108 cards in a deck of UNO cards.

The volume expansion of freezing water is roughly 108%.

108 degrees Fahrenheit is when the human body's organs began to fail from overheating.

In the TV show "Lost", the main characters must enter the numbers 4 8 15 16 23 42 into a mysterious computer every 108 minutes. (Note: 4+8+15+16+23+42 = 108).

281st Day of the Year.

281 = 28 + 81 + 21 + 128 + (28-1) + (8/(2*1))

281 = 18 + 82 + 12 + (2*(8-1) - 1)^2

281 = 28*1 + 2*81 + (2+81) + 8

281 = 18*2 + 1*82 + 21*8 + (2-8+1)

281 = (2+8+1)*(2*8*1) + 82 + (18-2) - (2-8-1)

281 = 182 + 128 - (28+1)

281 = 182 + (81-2) + (18+2)

281^2 starts with 789 (run of three consecutive numbers). (Are there other numbers that display this property?)

281 = (3!)^3 + (2!)^6 + (1!)^9

281 is a prime. 2+8+1 = 11 is also prime. 1+1 = 2 is also prime. (What are other numbers with this property?)

The sum of the first 14 primes is 281.

281, 2081, and 2801 are all prime. (What other numbers share this property?)

The first 281 digits of pi produce a prime number. (What's the next number with this property?)

281 = 9*8 + 7*6 + 5*4 + 3*2 + 1 + 2*3 + 4*5 + 6*7 + 8*9.

There are 281 primes between 44 and 44^2.

281 is prime.

2^3 + 8^3 + 1^3 = 521 is prime. (What's another number with this property?)

281 is the smallest prime whose digit product is 16.

281 = 29 + 31 + 37 + 41 + 43 + 47 + 53 (7 consecutive primes).

281 is the largest prime such that (1! + 2! + 3! + ... + 280! + 281!) - 2 is prime. (This is the largest prime with this property.)

###
October 9th (282nd Day of the Non-Leap Year)

Oct 9th, 10/9.

109 = 91 + 19 + (10-9)

109 = 10 + 9*((10-9)+10)

109 = 10 + 19 - (1-0-9)*(1+0+9)

109 = 1+2+3+4+5+6+7+8+9 + (9-1)^(10-9+10-9)

109*91 = 9919 (only using 9 and 1, excl--0) (Do other numbers have this property?)

109 is the first number such that |first digit - second digit + third digit - fourth digit + ... | = 10. (What's the next number with this property?)

109^2 + 110^2 is prime. (What's another number with this property?)

There are 109 primes less than 600. (109 being one of them)

90*109 + 19 is prime. (Note: Only 1,0,9 are used in this expression.) (Are there other numbers with a similar property?)

109 is a prime formed by the concatenation of two consecutive numbers, 10 and 9. (What's the next prime like this?)

109 is the smallest number that is palindromic in base 5 (414) and base 9 (131).

Below 109 degrees Fahrenheit, carbon dioxide becomes a solid (dry ice).

109^2 = 11881 and 118-8-1 = 109.

109 = 1*2 + 3*4 + 5*6 + 7*8 + 9

The Sun's diameter is a bit over 109 times the diameter of the Earth.

109 = prime(prime(1+0+9)) where prime(n) denotes the nth prime number.

109 is the smallest number that has more distinct digits than its square. (What's the next number with the same property?)

282nd Day of the Year.

282 = 28 + 82 + 22 + (28+2) + (2+82) + (2-8)^2

282 = 228 + (8-2)*(8+2/2)

282 = (2+8+2)*(2*8*2) - (28*2) - (28-2) - (2+8)*2

282 = 22 + 2*82 - (2-8-2)*(2+8+2)

282 = 28*2 + 2*82 + (82-28) - (2-8-2)

282 = 22*8 - (2-82) + (28-2)

282 = (3!)^3 + (2!)^6 + (1!)^9 + (0!)^12

There are no primes between 2820 and 2829. (What other numbers exhibit this property?)

282^6 + 282^5 + 282^4 + 282^3 + 282^2 + 282 + 1 is prime. (What's another number that satisfies this property?)

282^2 = 79524.

79524//282//1 = 795242821 is prime. (Are there other numbers with this property?)

10^282 + 100^282 - 1 is prime. (What's the next number with the same property?)

The sum of the divisors 282 is a square (576). (What are other numbers with this property?)

282 = 2*3*47 (2 < 3 < 4 < 7). (What's another number with the same property?)

###
October 10th (283rd Day of the Non-Leap Year)

Oct 10th, 10/10.

1010 = 101 + 10 + 11 + (10-1)*(101) - (10+10+10/10)

1010 = 10*10 + (101-10)*(10)

1010 = (10+1)*(101-10) + (10-1)

1010 = (101)*(10)

1010!!!!! + 1 is prime. (What's the next number with this property?)

There are 1010 primes less than 10,000 that do not contain the digit 0.

283rd Day of the Year.

283 = 28 + 83 + 23 + (2*83) - (8+3^2)

283 = 38 + 82 + 32 + (2*8*3) + 83

283 = (2+8+3)*(2*8*3) - 328 - (2+8+3)

283 = 2*83 + 28*3 + (2+8*3) + (2+8-3)

283 = (28+3) + (2+83) + (38+2) + (3+82) + (32+8) + (8-2*3)

283 = 238 - (2-8-3)*(2+8+3) - (82-3) + (2+8-3)

283 is the 61st prime. (Note: 283 and 61 are both prime.) (What's another number with this property?)

283 is prime and 2+8+3 = 13 is prime. (What are other primes with the same property?)

There are 283 primes less than 43^2.

283 = 3^3 + 4^4

283 is a concatenation of two primes (2//83). (What's the next prime to do this?)

283 is prime as 4! + 283 is also prime. (What's another prime with the same property?)

2^3 + 8^3 + 3^3 is prime. (What other primes share this property?)

283 in base 11 is 238 (uses same digits). (Do other numbers have this property?)

283 in base 9 = 344

283 in base 8 = 433 (flipped digits) (Are there other numbers that share the same property?)

###
October 11th (284th Day of the Non-Leap Year)

Oct 11th, 10/11.

1011 = 111 + 101 + 110 + 11 + 11*(10+11) + (1+0+1+1)*(110) + 10*11 + (10/(1+1) + (1+1))

1011 = 1110 - (10-1)*11

1011 = 1101 - (10-1)*10

1011 = 101*11 - 10*(11-1)

1011 = (1+0+1+1)*(111 + 101 + 110 + 11) + (1+0+11)

1011 = 10^(1+0+1+1) + 10^1 + 10^0

1011 = (1+0+1+1)*(10+11)*(10+11) - 101*(1+0+1+1) - (10-1)

1011*1101 = 1113111 (Note: 1+0+1+1 = 3). This number is also four different palindromes centered at 3 (3, 131, 11311, 1113111). (What are other numbers with this property?)

1011 is 11 (prime) in base 2 (Note: 11 is a substring of 1011). (Do other numbers have this property?)

1011 is divisible by 1+0+1+1. (What's the next number like this?)

101^2 + 101 + 1 is prime. (What's another number satisfying this property?)

1011^2 = 1022121

1101^2 = 1212201 (reverse the digits) (What are other numbers with the same property?)

1011 in base 3 is 1101110. (Note: 3 = 1+0+1+1 and 1011 is contained in the base 3 representation.) (Do other numbers have these properties?)

1011 in bases 6,7,8 and 9 all end in a 3. (Are there other numbers like this?)

284th Day of the Year.

284 = 28 + 84 + (2+84) + (82+4)

284 = 48 + 82 + 42 + 24 + (2-8+4) +(2+84)

284 = (28+4) + (2+84) + (48+2) + (4+82) - (2-8*4)

284 = 2*84 + 28*4 + (2+8/4)

284 = 248 + (2+8-4)^2

284 = 48*2 + 4*82 - (2*8*4)*(2-8+4) + 8+4

284 = (2+8+4)*(2+8+4) + (82+4) + 2

284 = (2*8*4)*(2-8-4) + 482 + 428 + (2+8+4)

The sum of divisors of 284 is 220.

The sum of divisors of 220 is 284. (What's another number with the same property?)

The sum of divisors of 284 = 220.

(Note: 284 - 2*8*4 = 220). (Do other numbers share this property?)

284*482 = 136888 (ends in repdigit). (Does this property hold for other numbers?)

284*(2+8+4) = 3976.

284*(2*8*4) = 18176. (Both end in same two digits.) (Are there others with the same property?)

2^2 + 8^2 + 4^2 = 84 (substring of 284). (Are there other numbers with this property?)

284^2 + 1 is prime. (What other numbers have this property?)

284 = 233 + 51 (51st prime plus 51. 233 is also a Fibonacci number.)

There are 284 primes between 31^3 and 32^3.

284^2 + 285^2 + 286^2 + 287^2 + 288^2 + 289^2 is prime. (What's the next initial number to satisfy this property?)

284!!!!! - 1 is prime. (What's the next number that has the same property?)

284, 285 and 286 are products of exactly 3 primes. (What's the next set of consecutive numbers this occurs for?)

284 degrees Celsius is the flash point of paper.

###
October 12th (285th Day of the Non-Leap Year)

Oct 12th, 10/12.

1012 = 10^(1+2) + (10+1*2)

1012 = 112 + 101 + 110 + 201 + 211 + 102 + 10*12 + (10+12)*(1+0-1+2) + 11

1012 = 1201 - (21-(1+0+1+2))^2 + 10^(1*2)

1012 = 211*(1+0+1+2) + 112 + (12-10)*(21-(2+1)) + 10*1*2

1012 = 12*12 + 21*21 + 10*10 + 1*1 + 10*12 + 21*10 - (1+0+1+2)

1012 = (112 + 121)*(1+0+1+2) + (21+12+10+11+20) + (10-1-2) - 1

1012 = 2^10 - 12

1012*1012 = 1024144 (concatenation of 32^2 and 12^2, 1024//144) (Are there others that share this property?)

1012*2101 = 2126212 (three 3-dig palindromes sharing one digit. (212,262,212))

1012 = 2*2*11*23

1012*(2+2+11+23) = 38456 (3,4,5,6 in order) (Do other numbers have this property?)

The number 4 appears 1012 times in the first 10^4 digits of pi. (Note: 4 and 10^4...)

285th Day of the Year.

285 = 28 + 85 + 25 + (2+85) + (2+58)

285 = 58 + 82 + 52 + (5+82) + 2*(8-5)

285 = 258 + (2-8+5) + 28

285 = 2*85 + 28*5 + 25

285 = (2+8+5)*(2*8*5) - 852 - 58 - (2+8-5)

285 = 58*2 + 5*82 - 258 + (25-8)

285 + 2*8*5 = 365 (365 days a year). (Does this happen for 366 as well?)

285 = 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2.

285 in base 7 is a repdigit (555). (What other numbers are repdigits (>3) in other bases?)

###
October 13th (286th Day of the Non-Leap Year)

Oct 13th, 10/13.

1013 = 301 + 310 + 311 + (101-10)

1013 = 101*3 + 10*13 + 31*10 + 113 + 131 + (10+13) + (1+0-1+3)

1013 = (1+0+1+3)*(10*13) + 311 + 13*(1+3)

1013 = 13*31 + 11*13 + 31*10 + 103 + (10+13) + 31

1013 = (1+3)^(1+0+1+3) - 3^(1+0+1)

1013 = 10^(1*3) + 10 + (1+0-1+3)

1013^3 contains only odd digits (excluding zeroes). (Are there other numbers like this?)

100^1013 - 99^1013 is prime. (What's the next number with this property?)

1013 is prime and eliminating any digit keeps the number prime (101,113,103,13) (What other numbers display the same property?)

1013 = 23^2 + (23-1)^2 (Note: 10+13 = 23).

The next prime is 1019. 10131019 is prime. (1013//1019) (When is the next time this will happen?)

1013 is prime. 1+0+1+3 = 5 is prime. 10+13 = 23 is prime.

1013 is prime.

1^2 + 0^2 + 1^2 + 3^2 = 11 is prime.

1^3 + 0^3 + 1^3 + 3^3 = 29 is prime.

1^4 + 0^4 + 1^4 + 3^4 = 83 is prime. (Do other primes have this property?)

If 1013 = p, 2p3p5p7 (that is, 2101331013510137) is prime. (What other primes satisfy this?)

There are 1013 non-attacking knights on a 45 by 45 board.

Putting any digit inside of 1013 will make the number composite. (What's the next number that has this property?)

Replacing any digit in 1013 with 9 still results in a prime. (What other numbers does this work for?)

There are 1013 ways ten people can line up such that only one person has a taller person in front of him.

2^1013 + 3^1013 + 5^1013 + 7^1013 is prime. (What's the next prime like this?)

The average sea-level pressure on Earth is 1013 millibars.

286th Day of the Year.

286 = 28 + 86 + 26 + 68*2 + (2*8-6)

286 = 68 + 82 + 62 + (86+(2-8-6))

286 = 28*6 + 2*86 - (62-8)

286 = 6*82 + 68*2 - 268 - (82-6) + (6-8/2)

286 = (2+8+6)^2 + (28+6) - (2+8-6)

286 = (28+6) + (2+86) + (68+2) + (6+82) - (2-8)

286 = (2*8*6)*(2+8-6) - (2+86) - (2*8-6)

286 = (26-8)*(2+8+6) - 2

286*(2*8*6) = 27456 (alternates even/odd digit) (What other numbers share this property?)

68*2 + 6*82 = 628.

286*(2+8+6) = 4576 (4 consecutive numbers). (Do other numbers share the same property?)

286*(28*6) = 48048 (ends with same two digits it begins with). (Are there other numbers with this property?)

286 = 2*11*13

2+11+13 = 26.

2+(1+1)+(1+3)=8. (Both these numbers make up '286'.) (Do other numbers have this property?)

286 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2.

286!!!! + 1 is prime. (What's another number with this property?)

286! - 285! - 1 is prime. (What's the next number with this property?)

286 = 7^3 - 7^2 - 7^1 - 1.

###
October 14th (287th Day of the Non-Leap Year)

Oct 14th, 10/14.

1014 = 411 + 401 - 101*(1+0+1-4)

1014 = 101*4 + 10*14 + 410 + 4*(1+0+14)

1014 = 14*41 + 4*(110)

1014 = (1+0+1+4)*101 - (1-0-1-4)*101 - (10-14)

1014 = (10+14)^(4-1-0-1) + 410 + (10+14) - (10-14)

1014 = (1+0+1+4)*(14-0-1)^(4-1-0-1)

The number 9 appears 1014 times in the first 10000 digits of pi.

1014^64 + 1015^64 is prime. (What's the next number with this property?)

The divisors of 1014 together use up the digits 0-9 at least once. (Is there a number whose divisors use 0-9 only once?)

1014/(1+0+1+4) is a square. (What's the next number with the same property?)

1014^2 - 1014 - 1 is prime. (What's another number with this property?)

1014 is base 5 uses all possible digits only one (0,1,2,3,4) (What other numbers satisfy this property?)

287th Day of the Year.

287 = 28 + 87 + 27 + (2+87) + 2*(2+8-7)

287 = 78 + 82 + 72 + 27 + 28

287 = 28*7 + 2*87 - (78+2) - (7-8-2)

287 = 7*82 + 78*2 - 278 - 87 - 82 + 28/7

287 = (2+87) + (28+7) + (78+2) + (7+82) - (2*7-8)

287 = 278 + (2*8-7)

287 = 2*8*7 + (87+2) + (87-2) + (2-8+7)

287 = 27*8 + (2+8+7) + 782 - 728

Inserting a 1 or a 0 between any digits of 287 will result in a prime. (What other numbers, if any, have one or both of these properties?)

287^8 + 288^8 is prime.

11^287 + 2 is prime. (Only one (known) number higher than 287 has this property. What is it?)

287^2 + 287 + 1 is prime.

287^32 + 288^32 is prime.

(What other numbers have one of these properties? Do other numbers have at least 2,3,4 of these properties?)

###
October 15th (288th Day of the Non-Leap Year)

Oct 15th, 10/15.

1015 = 511 + 501 + (5-1-0-1)

1015 = 510 + 101*5

1015 = 501 + 115 + 151 + 150 + 105 - (1+0+1+5)

1015 = (1+0+1+5)*(10*15) - (51-10) + (1+0+1*5)

1015 = (1+0+1+5)^(5-1-0-1) + 510 + (51+10) + 101

1015 = 1510 - (1+0+1+5)*(51+15) + (1-0-1-5)*(1+0+1+5) + 10/(1*5)

1015^3 = 1045678375 (contains '45678' run). (What numbers display a similar property? Run of 6, 7, 8 numbers?)

1015*5101 contains 1,5, and 1+0+1+5 (7) as its digits. (Do other numbers share this property?)

1015 = 5*7*29

1015*(5 + 7 + (2+9)) = 23345 (2 â‰¤ 3 â‰¤ 3 â‰¤ 4 â‰¤ 5). (Are there others with this property?)

1015 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2

1015 in binary only has one 0 in its decimal representation. (Where is it?)

288th Day of the Year.

288 = 28 + 88 + 82 + (2+88)

288 = 28*8 + 2*88 - (28+8) - (82-8) - 2

288 = (28+8) + (2+88) + (88+2) + (8+82) - (2+8+8)

288 = (2+8+8)^2 + (28+8)

288 = (2*8*8)*(2+8/8) - (2+88) + (2-8-8) + 8

2^4 + 8^4 + 8^4 = 8208 (same digits excl 0) (What other numbers, if any, have the same property?)

288*882 = 25016 (starts with 5^2, ends with 4^2)

288*(2*8*8) = 36864 (starts with 6^2, ends with 8^2)

(Are there other numbers like this?)

288 = 2*2*2*2*2*3*3

288*2222233 = 640003104 (run of 3 zeroes).

6+4+0+0+0+3+1+0+4 = 18 = 2+8+8

(Do other numbers have both these properties? What numbers have just one of the properties?)

288 = 4!*3!*2!*1!*0!

288^4 + 1 is prime.

288^8 + 1 is prime.

288^16 + 1 is prime.

(Note the exponents are of the form 2^n. What other numbers have 1, 2, or all 3 of these properties?)

288 is divisible by 2,8, and 8. (What's a nontrivial number (no repdigits or 1) with this property?)

288 = 4^4 + 3^3 + 2^2 + 1^1

288^2 + 288 - 1 and 288^2 + 288 + 1 are twin primes. (What's another number with this property?)

288 in base 3, 4, 6, and 12 all end in 200.

Bases 3 and 6 both end in 1200.

(What other numbers have one or both of these properties?)

288 in any base < 17 will have only numbers, no letters. (What's the next number with this property?)

###
October 16th (289th Day of the Non-Leap Year)

Oct 16th, 10/16.

1016 = 611 + 110 + 106 + 161 + (10+16) - 10/(1-6)

1016 = 601 + 116 + 101 + 160 + (1*0+1*6) + (1+0+1)*16

1016 = 1610 - 611 + (1+0+16)

1016 = 116*(1+0+1+6) + (61+10) + (1+0+16)

1016 = (61-16 + (6-1)) - 161*(1-0-1-6)

1016 = ((1+0+1+6)+(1-0-1-6))^10 - ((1+0+1+6)+(1-0-1-6))^(6/(1+0+1))

6101*6101 = 37222201 (four 2s in a row). (What other numbers have a similar property? Five 2s, 3s, 4s,...?)

1016*(10+16) = 26416 (concatenation of 2^1, 2^6, 2^4) (Are there other numbers that display the same property?)

1016 + (1+0+1+6) = 1024 = 2^10. (What's the next number with the same property?)

1016 = |2^3 - 4^5|

There are 1016 primes between 47^pi and 48^pi.

289th Day of the Year.

289 = 28 + 89 + 29 + 2*8*9 - (2+8-9)

289 = 98 + 82 + 92 + (2+8+9) + (2*(8-9))

289 = 28*9 + 2*89 - 29*8 + (2+89)

289 = 98*2 + 9*82 - 829 + (28+9) + (2+89) - 28*(2*(8-9))

289 = (2+8+9)*(2+8+9) - (82-9) + (8+2-9)

289 = 2*8*9 + (89-2) + (82-9) + (2-8-9)

289 = (28+9) + (2+89) + (9+82) + (98+2) - 29 - (2+8-9)

289 = [(2*9-8) + (2*8-9)]^2

289^3 = 24137569 (each number is unique). (Are there numbers whose cube (or square or 4th power) give all digits 0-9 once?)

(8+9)^2 = 289. (Do other numbers share this property?)

289 = 17^2.

2*8*9 = 12^2. (Are there others with this property?)

289!!!!! + 1 is prime. (What's another number with this property?)

289^2 = 83521.

2+8+9 = 8+3+5+2+1. (What's the next number with this property?)

289 = 17^2.

289 in base 4 = 10201 = 101^2.

289 in base 8 = 441 = 21^2.

289 in base 14 = 169 = 13^2.

289 in base 15 = 144 = 12^2.

289 in base 16 = 121 = 11^2.

289 in base 17 = 100 = 10^2.

289 in base 36 = 81 = 9^2.

(What other base 10 squares share any of these properties?)

###
October 17th (290th Day of the Non-Leap Year)

Oct 17th, 10/17.

1017 = 711 + 171 + 117 + (1+0+17)

1017 = 101*7 + 170 + (1+0+1)*70

1017 = 70*11 + 107 + 10*17 - (10+17) - (10-1*7)

1017 = 1701 - 711 + (10+17)

1017 = 71*(10+1) + 171 + (71-10) + (10+1-7)

1017 = 71*(1+0+1+7) + 170 + 107 + 101

1017^2 = 1034289 (all different digits). (Note that 1017 does not have all distinct digits. What's the next number displaying both properties here?)

All of the proper divisors of 1017 only use 1,3, or 9 (Note: 1 = 3^0, 3 = 3^1, 9 = 3^2) (Do other numbers display similar properties?)

1017 is the last of a run of 4 numbers that are divisible by their sum of digits (1017/9, 1016/8, 1015/7, 1014/6).

43^1017 - 42 (1662-digit number) is a "probable prime". (1017 is the highest known number that returns a prime, The next highest number has to be more than 3000.)

290th Day of the Year.

290 = 90 + 20 + 2*90

290 = 29 + 90 + 92 +20 + 90/2 - 2*(2-9-0)

290 = 209 + 9*2

290 = 9*20 - (2+9+0)*(2-9-0) + 29 + (2+9) + (2-9)

(2+9) - 2*9 = (2-9)

There are 290 primes less than 1900.

290^6 + 290^5 + 290^4 + 290^3 + 290^2 + 290 + 1 is prime. (What's the next number with this property?)

290^8 + 1 is prime. (What's the next number with the same property?)

290^2 + 290 - 1 and 290^2 + 290 + 1 are twin primes. (What's another number with this property?)

290 is the lowest number such that 6*290 + 1, 6*290 + 7, 6*290 + 13 and 6*290 + 19 are 4 consecutive primes. (What's the second number to display this property?)

If you writing 290 in bases 3, 4, 6, and 12 the decimal representations all end in '202'. (What other numbers have a similar property?)

290 = 67 + 71 + 73 + 79

###
October 18th (291st Day of the Non-Leap Year)

Oct 18th, 10/18.

1018 = 101*8 + 181 + 10 + (1+0+18)

1018 = 8*110 + 118 + 18 + (10-1*8)

1018 = 811 + 180 + 10 - (1-0-18)

1018 = 10*18 + 801 + (10+18) + (1+0+1*8)

1018 = 1801 - 811 + (10+18)

1018 = 18*81 - 110*(8/(1+0+1))

1018 = 101*(1+0+1+8) + (1*0+1*8)

1018 = (1+0+1+8)*108 - (81-10) + (1*0+1+8)

1018 = 118 - (1-0-1-8)*108 + (10+18) + (8-1-0-1)

1018^16 + 1 is prime. (What's the next number with this property?)

The sum of the proper divisors of 1018 is a cube (8^3). (What's the next number displaying this property?)

2^1018 - 57 is prime. (What's another number with this property?)

If 1018 = n, then the concatenation 9nn9 is prime. (Meaning, 9101810189 is prime.) (What's another number like this?)

1018^256 + 1019^256 is prime. (What's the next number with the same property?)

1018^5 contains all digits 0-9 at least once.

291st Day of the Year.

291 = 29 + 91 + 21 + 129 + (2+9*1) + (2+9-1)

291 = 19 + 92 + 12 + 192 - 2*(2+9+1)

291 = 192 + 129 - (29+1)

291 = 2*91 + 29*1 + (92-1) + (2+9*1)

291 = 1*92 + 19*2 + 129 + 2^((9+1)/2)

291 = 29*91 - 912 - 921 - 219 - 129 - 192 + (29-1) - 9/(2+1)

291 = (2*9-1)^2 + 2

291 = (2*9*1)*(2+9+1) + 92*1 - (2*9-1)

291 = 239 + 52 (52nd prime plus 52)

10^291 + 13 is prime. (What's another number with a similar property?)

The sum of proper divisors of 291 is prime (101). (What's another number like this?)

There are 291 primes of the form 1 + n^16 (n is less than or equal to 10^4).

10^291 - 11 is prime. (What's the next number like this?)

10^54 - 291 is prime (largest prime < 10^54).

2^291 - 19 is prime (Note: 2,1,9 are in 291). (Are there other numbers with this property?)

292^11 + 291^11 is prime. (What's the next number displaying this property?)

291! - 290! - 1 is prime. (What's another number with the same property?)

291^2 + 292^2 + 293^2 + 294^2 + 295^2 + 296^2 is prime. (What other numbers have this property?)

If 291 = n, n1n and 1n1 are prime. (Ex. 2911291 and 12911 are prime). (What's the next number with this property?)

291 = 97*3 (97 > 3, 9 > 7 > 3). (Are there other numbers with the same property?)

###
October 19th (292nd Day of the Non-Leap Year)

Oct 19th, 10/19.

1019 = 911 + (109-1)

1019 = 901 + 109 + (1*0+1*9)

1019 = 910 + 109

1019 = 101*9 + 110

1019 = 110*9 + (10+19)

1019 = (10+19)^(10+1-9) + (9-1-0-1)*(1+0+1+9) - (1+0+1+9)*(1-0-1-9) + (10+1-9)

1019 = 91*10 + 10*19 - 9^(1+0+1)

1019 = 91*19 - (9-1-0-1)*101 - 9^(1/(1+0+1))

1019^3 is a ten digit number with 9 of its digits represented by only 4 distinct digits.

9101*9101 begins "828282..."

1019*91 = 92729 (palindrome).

1019 is prime.

Deleting any nonzero digit in it keeps it prime.

2*3*5*7*11*13*17*...*1019 + 1 is prime.

There are 1019 primes less than e^9 (8103).

1013 was the prime before this. 10131019 is prime. (concatenation of 1013//1019)

1019 is a prime of the form 4^n - n (n = 5 in this case).

1019 written in base 2 ends in a 1.

1019 written in base 3 ends in a 2.

1019 written in base 4 ends in a 3.

1019 written in base 5 ends in a 4.

1019 written in base 6 ends in a 5.

292nd Day of the Year.

292 = 29 + 92 + 22 + 22*9 - (2*9*2) - (2+9+2)

292 = 29*2 + 2*92 + (29+2) + (2+9+2) + (9-2-2/2)

292 = (29+2) + (2+92) + (22+9) + 22*9 - (29+2)*2

292 = 229 + 9*(9-2)

292 = 2*9*2 + 2^(9-2/2)

292 = 2*2*73

2+2+73 = 77 (also a palindrome).

292*77 = 22484 (concatenation of two palindromes 22//484).

292 - (2*9*2) = 16^2 = 2^8 = 4^4

292*29, 292*22 and 292*92 both have only even digits. (29,22,92 are the different two-digit numbers created using 292)

292 = 73*2*2

73 â‰¥ 2 â‰¥ 2 and 7 â‰¥ 3 â‰¥ 2 â‰¥ 2.

There are 292 primes between 45 and 45^2.

There are 292 squares less than 44^3.

292^5 - 291^5 is prime.

There are 292 primes of the form n^16 + 1 less than 10^64.

292 is also a palindrome in base 7 (565) and base 8 (444).

292 written in base 2 is 100100100 (concatenation 100//100//100).

The continued fraction of pi is [3; 7, 15, 1, 292, 1, 1, 1, 2, ...]

###
October 20th (293rd Day of the Non-Leap Year)

Oct 20th, 10/20.

1020 = 2^10 - 2^((1+0)*(2+0))

1020 = 201 + 200 + 210 + 120 + 102 + 100 + (10+20)*(1+0+2+0) - (1+0+2+0)

1020 = 201*10 - 1002 + 12

1020 = 4*(4^4 - 1)

There are 1020 ways to place 2 non-attacking kings on a 7 by 7 board.

1020!!!!!! + 1 is prime. (What's the next number to display this property?)

3*(10^1020) + 1 is prime. (What's another number with the same property?)

1 + 1020 + 1020^3 + 1020^5 + ... + 1020^45 + 1020^47 is prime. (What other numbers have this property?)

1020 is the smallest 4 digit number with 4 prime divisors. (What's the next number with 4 distinct prime divisors?)

1020 written in bases 2,3,4,5,6 all end in a 0 (1111111100, 1101210, 33330, 13040, 4420 respectively). (Are there other numbers where this happens?)

The first digit of 1020 (1) is divisible by 1.

The first 2 digits of 1020 (10) are divisible by 2.

The first 3 digits of 1020 (102) are divisible by 3.

The first 4 digits of 1020 (1020) are divisible by 4.

(What's the first 5 digit polydivisible number?)

293rd Day of the Year.

293 = 29 + 93 + 23 + (2+93) + (39+2) + 2*(9-3)

293 = 39 + 92 + 32 + (92-3) + (32+9)

293 = (2*9*3)*(2+9+3) - 392 - 2*9*3 - (2^3+9)

293 = 239 + 2*3*9

293 = 329 - (9-3)^2

293 = 2*93 +29*3 + (2+9-3) + 2*(9-3)

293 = 3*92 + 39*2 - 29 - 32

293 = (29+3) + (2+93) + (3+92) + (39+2) + 2*9*3 - 2*(9+3)

293^3 contains all prime digits (excl 1). (Are there other primes or numbers in general like this?)

293 is prime.

293^2 + 293 + 1 is prime.

(10^293 + 1)/11 is prime.

(What's the next prime with one or both of these properties?)

293 is of the form p^2 + 4 for prime p (p = 17).

Also, 293^2 + 4 is prime.

(Another way of seeing this...)

17 is prime.

17^2 + 4 = 293 is prime.

293^2 + 4 = 85853 is prime.

(What other primes have these properties? Do other primes keep going?)

The string '24' does not appear in pi until the 293rd (and 294th) digit. (When does the string '293' first appear?)

293 is prime.

293 + 2*9*3 = 347 is prime.

293 - 2*9*3 = 239 is prime.

293 + (2+9+3) = 307 is prime.

(What other primes have at least one of these properties? Other primes with all three?)

293 is not a palindrome in bases 2,3,4,...291. (What's the next number with this property?)

###
October 21st (294th Day of the Non-Leap Year)

Oct 21st, 10/21.

1021 = 210 + 201 + 211 + 102 + 112 + 121 + 120 - (1+0+2+1)*(21-10 + (1+0+2+1))

1021 = 1201 - 211 + (10+21)

1021 = 10^(2+1) + (10*2+1)

1021 = (12*201 - 102*21)*(1+0+2+1) - (10+21) - (12+1) - (12+1+2)

1021^2 = 1042441 (take away 4s, get 1021). (Are there other numbers that have this property?)

1021*1201 = 1226221 (palindrome) (What's the next number that has this property?)

1021 = 2^10 - 2^2 + 2^1 (Note the exponents: 10,2,1)

1021^2 = 1042441

1201^2 = 1442401 (reverse of 1021^2)

(What's the next number displaying this property?)

1021 is prime.

1201 is prime. (What are other numbers with this property?)

2*3*5*7*11*13*...*1019*1021 + 1 is prime. (What is the next number to do this? Hint: Over 2500!)

1021 is the smallest 4-digit number with the smallest digit sum (coincidentally, it's 4). (What's the next smallest 4- digit prime with the next smallest digit sum?)

The number 2 appears 1021 times in the first 10000 digits of pi.

The number 6 appears 1021 times in the first 10000 digits of pi.

DigitSum(1021^1) = 4^1.

DigitSum(1021^2) = 4^2.

(Is there another number with this property?)

294th Day of the Year.

294 = 29 + 94 + 24 + 2*94 - (92/4) - (9*4/2)

294 = 49 + 92 + 42 + 2*9*4 + (2+9+4) + (29-4) - (9-2*4)

294 = 429 - 9*(2+9+4)

294 = 2*94 + 29*4 - 2*(9-4)

294 = 4*92 + 49*2 - 249 + 2*9*4 - (2+9+4)/(2-9+4)

294 = 24*9 + 9*42 - 249 - (2+49)

294 = (29+4) + (2+94) + (4+92) + (49+2) + (2+9+4) + (2-9+4)

294 = 29*4 - (2-9-4)*(2+9+4) + (9+4)

294 = (2+9-4)^(9-4-2) - 49

294 = 2*3*7*7 (2 â‰¤ 3 â‰¤ 7 â‰¤ 7). (What's another number sharing this property?)

There are no primes between 2940 and 2949. (What is another number with the same property?)

294 = 241 + 53 (53rd prime plus 53).

294^6 + 294^5 + 294^4 + 294^3 + 294^2 + 294 + 1 is prime. (What's the next number with this property?)

There are 294 groups with an order less than or equal to 56.

294 + 2*9*4 = 366 (366 Days in a Leap Year). (Do other numbers have this property?)

294 - 2*9*4 = 222 (repdigit).

294 - 49 - 92 - 42 = 111 (repdigit). (Are there other numbers with similar properties?)

294 = 7^2 + 8^2 + 9^2 + 10^2

294^2 + 295^2 + 296^2 is prime. (What's another number with this property?)

###
October 22nd (295th Day of the Non-Leap Year)

Oct 22nd, 10/22.

1022 = 221 + 202 + 210 + 120 + 212 + (10+22) + 22 - (1-0-2-2)

1022 = 102*21 - 1120

1022 = (10*22)*(1+0+2+2) - 102 + (10+22) - 2^(2+0+1)

1022 = (10+22)*(22-10) + (10+22)*(22-1) - (10+22) - (1+0+2/2)

1022 = (1*0+2+2)^(1+0+2+2) - (1+0+2/2)

1022*2201 = 2249422 (palindrome).

1022^2 = 1044484.

2201^2 = 4844401. (reverse of 1044484)

1022^9 + 1022^7 + 1022^5 + 1022^3 + 1022 + 1 is prime.

1022 is a divisor of 299792458, the speed of light in meters per second.

The sum of divisors of 1022 is 1776 (year Declaration of Independence was ratified).

(Also, 1022 written in base 8 is 1776).

1022 written in base 5 uses all the digits 0 through 4 (13042).

295th Day of the Year.

295 = 29 + 95 + 25 + 2*95 - (52-9) - (2*5-9)

295 = 59 + 92 + 52 + (95-2) + (9-2*5)

295 = 2*95 + 29*5 - (59-2) + (2+9+5) + (2*5-9)

295 = 259 + (2+9-5)^(9-5-2)

295 = 5*92 + 59*2 - 259 - (29-5)

295 = 25*9 + 52*9 - 259*(9-5-2) + (2*9*5) + (29-5) + (2+9-5)

295 = (2+95) + (29+5) + (59+2) + (5+92) + (2+9-5)

295^6 is the first power of 295 that has a 9 (excluding 295^1)

There are 295 primes less than 44^2.

If 295 = n, (1n^2+3)/4 is prime (Note the numbers used: 1 then 2 then 3 then 4)

There are 295 primes of the form 8n+1 less than 10000.

295 + 592 = 887. (not a palindrome)

887 + 788 = 1675. (not a palindrome)

1675 + 5761 = 7436 (not a palindrome)

....

295 is a (proposed) Lychrel number. That is, adding its reverse of digits will never generate a palindrome.

###
October 23rd (296th Day of the Non-Leap Year)

Oct 23rd, 10/23.

1023 = 321 + 310 + 312 - (10+23)*(1-0-2)*3 - 23 - (1-0-2-3)

1023 = 213 + 230 + 201 + 203 + 231 - (31+20) + (1-0-2-3)

1023 = (1+0+2+3)*(10*23) - 321 - (10+23) - (10+23)/(13-2)

1023 = 1320 - 301 - (1-0-2-3)

1023 = 32^(10/(2+3)) - (10-3^2)

1023 = 2013 - 1032 + (10+32)

1023 = 32*10 + 31*20 + (10+23)*(2-0-1)*3 - 23 + (1+0+2*3)

1023 = 312 + 10*23 + 13*20 + 213 + (1*0+2^3)

1023 = 2^10 - 3^2 + 2^3

1023 = 3*11*31

3+11+31 = 45 (uses next numbers that 1023 doesn't use). (Do other numbers have a similar property?)

1023 = 3*11*31 (only uses 3 and 1).

1023 = 2^9 + 2^8 + 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 1 = 2^10 - 1

1024^11 - 1023^11 is prime.

1023 is a palindrome in base 2 (1111111111), base 4 (33333), and base 32 (VV).

October 23rd is "Mole Day" in appreciation of Avogadro's number for the number of atoms/molecules per mole (6.022*10^23).

296th Day of the Year.

296 = 29 + 96 + 26 + 69*2 + (62-(2-9+6))/9

296 = 69 + 92 + 62 + (69-2) + 2*(9-6)

296 = 2*96 + 29*6 - (69+2) - (2-9+6)

296 = 69*2 + 6*92 - 269 - (2+9-6)^(2*9/6)

296 = 269 + 9*(6/2)

296 = 629 - 269 - ((9-6)!-2)^(6*2-9)

296 = (29+6) + (2+96) + (6+92) + (69+2) - (9-6/2)

296 = 26*9 + 62*9 - 629 + 69*2 - (2+9-6)

296 = (2+9+6)^((2+9-6)-(2*6-9)) + (69-(9-6/2))/9

296 = (2*9*6)*(2*9/6) - (29-6) - (2+9-6)

296^4 (10 digits) uses 3,4,5,6,7 only (and ends in 3456).

296 = 2*2*2*37 (2 â‰¤ 2 â‰¤ 2 â‰¤ 3 â‰¤ 7) (What's another number like this?)

296^4 + 1 is prime. (What's another number with this property?)

The sum of the first 296 primes is prime (263171).

18*296 + 1 and 19*296 + 1 are squares. (Are there other numbers with both properties?)

296^2 + 297^2 + 298^2 + 299^2 + 300^2 + 301^2 is prime. (What's the next number with this property?)

###
October 24th (297th Day of the Non-Leap Year)

Oct 24th, 10/24.

1024 = 4^(10/2)

1024 = 421 + 410 + 142 + (42+10) - (10/2-4)

1024 = 402 + 420 + 214 - (10-2+4)

1024 = 2014 - 1042 + (42+10)

1024 = 240 + 124 + 204 + 412 + (2*10+24)

1024^2 = 1048576 (all different digits, last digits are 4,5,6,7,8) (Is there a number with a similar property?)

1024 = 2^10 = 32^2 = 4^5 (the last in the form of n^(n+1))

1024 is the product of divisors of 16 (2^4).

There are 1024 ways to place 3 nonattacking queens on a 6 by 6 board.

1024 = 2^10 is the first four-digit number that is a power of two, and the first power of 2 with a nontrivial 0.

297th Day of the Year.

297 = 29 + 97 + 27 + ((2*9-7) - (2-9)/7)^2

297 = 79 + 92 + 72 + (72-9) - (2+7)

297 = 2*97 + 29*7 - ((9-7)*(7-2))^2

297 = 7*92 + 79*2 - 729 + 279 + (2-9*7) + (7*2-9 - (2-9)/7)

297 = 27*9 + 72*9 - 729 + 27*(7-2)

297 = 279 + (2+7+9)

297 = (29+7) + (2+97) + (7+92) + (79+2) - (2+9+7)

297 = 2*9*7 + 2*97 - (29-7) + (2-9)/7

297 = (9-7)*(2*9*7) + 9*(7-2)

297^3 has all 8 digits different. (Are there other numbers with this property?)

297*(2+9+7) = 5346 (uses 3,4,5,6). (Are there other numbers with a similar property?)

297 is a divisor of 999999.

297!! + 2^9 and 297!! + 2^7 are both primes (Note: 2^9 and 2^7 use the digits of 297). (Do other numbers have both these conditions?)

There are more primes contained in [297, 2*297] than [297, 297^2]. (What's the next number with the same property?)

297^5 - 296^5 is prime. (What's another number with the same property?)

There are only 5 groups of order 297. (Can you name one of these groups?)

2^297 + 17 is prime. (What's the next number with this property?)

###
October 25th (298th Day of the Non-Leap Year)

Oct 25th, 10/25.

1025 = 25*(52-10) - (10*2+5)

1025 = 521 + 502 + (5-1)/2

1025 = 501 + 520 + (10/5+2)

1025 = 251 + 205 + 150 + 152 + 215 + 52

1025 = 1250 - 25*(1+0+2+5+10/(2*5))

1025 = 102*25 - 1502 - (25-(5-2-0-1))

1025 is divisible by 25 (last two digits). (What's the next number with this property?)

1025*5201 ends in 1025. (What other numbers have this property?)

1025 = 5*5*41

(5 â‰¤ 5 â‰¤ 41 but 5 â‰¥ 5 â‰¥ 4 â‰¥ 1) (Do other numbers have this property?)

1026^11 - 1025^11 is prime (Note from the 1023rd day, 1024^11 - 1023^11 is also prime).

There are 1025 composites less than 35^2 (Note: 10+25 = 35).

1025 is a divisor of 2^20 - 1. (What are other divisors?

1025 degrees Celsius is prime in Fahrenheit (1877 F).

There are 1025 squares less than or equal to 2^20.

298th Day of the Year.

298 = 29 + 98 + 28 + 89*2 - (29+8) + 2*(9-8)

298 = 89 + 92 + 82 + (29-8) + (2-9-8) + (2-9+8)

298 = 2*98 + 29*8 - 2*9*8 + (9+8-2) - (2-(9-8))

298 = 89*2 + 8*92 - 829 + 2*9*8 + (89-2) - (28-9) + (2-9+8)

298 = 28*9 + 82*9 - 892 + 289 - (82+9) + (2*(9-8))

298 = (2+9+8)*(8*9-2) + (9+8/2)

298^2 begins with 888 (repdigit).

298*29 = 8642 (descending even digits, note: 29 is in 298) (What other numbers satisfy this property?)

298 = 149*2

1492*298 starts with 444 (repdigit, 1492 = 149//2)

298^5 - 297^5 is prime. (What's another number with this property?)

###
October 26th (299th Day of the Non-Leap Year)

Oct 26th, 10/26.

1026 = 621 + 201 + 206 - 6/(2+0+1)

1026 = 216 + 210 + 160 + 260 + 126 + (10+26) + (16+2)

1026 = 2601 - 2016 + 21^(6/2-0-1)

1026 = 102*26 - 1626

1026 = 2160 - 1062 - (62+10)

1026^3 has 3 zeroes. (Other numbers where N^m has m zeroes?)

The digit 1 appears 1026 times in the first 10000 digits of pi.

1026^64 + 1027^64 is prime.

299th Day of the Year.

299 = 29 + 99 + 92 - (2-9*9)

299 = 2*99 + 29*9 - 2*9*9 + 2*(9/9)

299 = 92*9 - (2+9/9)*(2*9*9) - (29+9) - (2+9/9)! + 2-(9/9)

299 = (2+9)*9 + 2*(9+9) + (2*9+9) + (2+9*9) + 9*(2+9/9)!

299 = (29+9) + (2+99) + (92+9) + (2+9+9) + (9+9/9) + 29

299 = 29 - (2-9-9)*(2+9+9) - (29+9/9) - (2+9+9)

299^3 begins with a 2 and ends with a 99. (Are there other numbers with this or a similar property?)

299^4 contains the reverse of 299 (992). (Do other numbers satisfy this?)

There are 299 semiprimes less than 1000.

299!! + 2^9 is prime (Note: Only 2 and 9 are being used).

###
October 27th (300th Day of the Non-Leap Year)

Oct 27th, 10/27.

1027 = 721 + 170 + 120 + (27-10) - (10-2-7)

1027 = 702 + 201 + 127 - (21/7)

1027 = 701 + 217 + 102 + (1*0*2+7)

1027 = 27*102 - 1727

1027 = 712 + 271 + (10+27) + 7

1027 = 102*(1+0+2+7) + 7

1027^2 has 1 zero.

1027^3 has 2 zeroes.

(number of zeroes is one less than exponent)

(Other numbers with this property?)

1027^2 has all different digits.

1027 = 13*79 (13 < 79 and 1 < 3 < 7 < 9). (What other numbers satisfy this?)

1027 = 13*79

1379*1027 ends in 233 (13th Fibonacci number--one of 1027's prime factors is 13).

3^1027, 3^1028, and 3^1029 all have the same number of digits. (What's the next number with this property?)

Concatenating 10, 10^2, and 10^3 with 1027 results in a prime (1010010001027 is prime).

1027 = 3 + 4^5

There are only 6 primes between 102700 and 102799. (Are there 5 primes between N*100 and N*100 + 99? 4? 0?)

300th Day of the Year.

300 = 3*30 + (30-3)*(300/30) - 30*(3-3^0)

300 = 30^3/(3*30)

300 = (30+3)*(3^(3-3^0)) + 3

300 = 3*(3*3 + 3^0)^(3-3^0)

300 = 2*2*3*5*5 (2 â‰¤ 2 â‰¤ 3 â‰¤ 5 â‰¤ 5). (What other numbers have this property?)

300 = 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24

300^2 + 1 is prime.

300^6 + 300^5 + 300^4 + 300^3 + 300^2 + 300 + 1 is prime.

300^64 + 1 is prime (300^64 has 130 zeroes).

300^8 + 301^8 is prime. (What's the next number with this property or any combination of the other prime properties above?)

300 = 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 = 149 + 151

300 is a perfect score in bowling (12 strikes).

In paintball, 300 ft/s is the maximum legal velocity of a paintball gun.

###
October 28th (301st Day of the Non-Leap Year)

Oct 28th, 10/28.

1028 = 820 + 208

1028 = 821 + 208 - 10/(2+8)

1028 = 812 + 201 - (1-0-2*8)

1028 = 102*8 + 10*28 - 82 + 2*(8-0-1)

1028 = 2^8 + (10*(8/2-1))^(20-18)

1028 = 2^10 + 8/(2+0*1)

1028 = 8^(2+0+1) + 281 + 218 + (1+0+2*8)

1028^2 = 1056784 (run of 5678). (Do other numbers have a similar property? Run of 1234? 2345? 34567? etc..)

1028 = 2*2*257 (prime factorization uses only prime digits). (What is another number with the same property?)

1028 = 2*2*257 (2 â‰¤ 2 â‰¤ 257 and 2 â‰¤ 2 â‰¤ 2 â‰¤ 5 â‰¤ 7). (Do others have this same property?)

There are 1028 prime less than 2^13 (Note: 2 and 13 are both prime).

1028 = 1^2 + 3 + 4^5

1028 = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2

1028^9 is a prime written backwards. (What's the next number with the same property?)

301st Day of the Year.

301 = 103 + 131 + 31 + 30 + 13 - (10-3)

301 = 31*13 + 103 - (3^0/1)

301 = 103*(3+0+1) - 130 + (3+0+1)^(3-0-1) + 3*(3+0+1) + (10-3)

301 = 10^(3-0-1) + 130 + 103 - (30 + (3-0-1))

301 = 13*30 - 103 + 13 + 3^0

301 = 13^(3-0-1) + 130 + (3-0-1)

301*103 = 31003 (only uses 3,0,1). (Are there others that satisfy this property?)

301 = 7*43 (Note: 7 = 4+3) (What's another number like this?)

301 = 7*43 (Note: 7 < 43 but 7 > 4 > 3). (Are there others with this property?)

301 = 43*7

437*301 = 131537 (all odd digits, 437 = 43//7). (Do other numbers display this property?)

301, 302, 303 are all semiprimes.

2^301 - 301^2 is prime. (What's the next number with this property?)

301 = 97 + 101 + 103 (3 consecutive primes)

301 is the number that YouTube freezes the "view" count on in order for validation to be carried out.

###
October 29th (302nd Day of the Non-Leap Year)

Oct 29th, 10/29.

1029 = 912 + 129 - (1+0+2+9)

1029 = 921 + 109 + (10-2-9)

1029 = 901 + 2^(9-2-0*1)

1029 = 210 + 291 + 219 + 290 + (29-10)

1029 = 2^10 + (92 - 29 - 2*9)/9

1029 = 2^9 + 291 + 129 + 109 - (1+0+2+9)

1029*(1+0+2+9) = 12348 (begins with 1234). (Does this happen with other runs? N*DigitSum(N) starts with 2345? 34567? etc.)

1029*91 = 93639 (palindrome) (Note: 91 the last and first digit concatenated in that order).

1029!!!!!!!! + 2 is prime. (What's another number with this same property?)

10^1029 + 11^1029 is divisible by 1029.

1^1029 + 2^1029 + 3^1029 + 4^1029 + 5^1029 + 6^1029 is divisible by 1029.

6^1029 + 7^1029 + 8^1029 is divisible by 1029.

9^1029 + 12^1029 is divisible by 1029.

3^1029 + 4^1029 + 5^1029 + 9^1029 is divisible by 1029.

(What numbers satisfy at least one of these properties? Two? Three? The numbers 1, 3, 7 and 9 satisfy all of these properties. What's the next number after 1029 that does?)

The sum of divisors of 1029 is a square (40^2).

302nd Day of the Year.

302 = 230 + 3*2*(20 - 2^3)

302 = 203 + 32 + 30 + 20 + 23 - 2*3

302 = 32*(2^3) + 32 + (20-2*3)

302 = 23*(3^2) + 3*23 - (3-0-2)

302 = 2*((3+0+2)^3 + (3+0+2)^0 + (3+0+2)^2)

302^3 contains all different digits (27543608). (What's another number displaying this property?)

302*23 and 302*32 have the same numbers (just not in the same place). (What other numbers, if any, are like this?)

302^1, 302^2, 302^3 all have only one zero at the tens digit. (Do other numbers satisfy this property?)

302^4 does not have a zero in it.

There are 302 squares less than 45^3. (Note: sqrt(45^3) rounds up to 302. The square root sign is for the fact that there are 302 squares.).

The concatenation of 302^2, 302 and 1 is prime. Hence, 302^2 = 91204 ==> 912043021 is prime. (What's a different number with the same property?)

302^64 + 303^64 is prime. (What's another number with this property?)

The sum of the divisors of 302 is 456 (run of consecutive numbers). (Are there others with this property?)

The sum of divisors of 302 is 456. 302 written in base 8 is 456. (Do other numbers have this property?)

302 written in base 9 is 365 (365 days in a non-leap year). (What numbers have 365 in a different base?)

###
October 30th (303rd Day of the Non-Leap Year)

Oct 30th, 10/30.

1030 = 1003 + (31-13) + 13 - (1+0+3+0)

1030 = 301 + 310 + 300 + 103 + (3-1)^(3+1)

1030 = 30*301 - (30-10)^3

1030 = 31^2 + 13^2 - 100

1030 = 13*31 + 301 + 310 + (3-1)^(3+1)

1030 = 1 + 2 + 3 + 4^5

1030 = 10^3 + 10*3

There are 1030 ways to place 3 nonattacking queens on a 3 by 13 board.

1030 is a four digit number with a digit sum of 4.

303rd Day of the Year.

303 = 330 - 3^3

303 = 30*3 + (3+0+3)*33 + 30/(3+3/3)

303 = 3*(30*3 + 30/3 + 3^0)

303 = (3+0+3)*3^3 + 30*(3+0+3) - (30+3) - (3+0+3)

303 = (3*3)*(30+3) + (3+0+3)

303^3 = 27818127 (concatentation of 27,81,81,27...all are powers of 3, the 3 is significant because this is 303 cubed).

303*(30+3) = 9999 (repdigit).

303 = 3*101

303*3101 = 939603

303*1013 = 306939 (digits in opposite order. Note: 1013 and 3101 are concatenations of 3 and 101.) (Do other numbers have this property? Does it have to do with the prime factorization or is there a general concept here?)

303^6 + 303^5 + 303^4 + 303^3 + 303^2 + 303^1 + 1 is prime. (What's the next number with this property?)

There are 303 primes less than 2000.

303^3 and 303^4 does not contain 0 or 3. (What other numbers have one or both of these properties?)

###
October 31st (304th Day of the Non-Leap Year)

Oct 31st, 10/31.

1031 = 1103 - 31 - (10+31)

1031 = 1301 - 310 + (110+10)/3

1031 = 310 + 311 + 301 + 130 - (31-10)

1031 = 103*13 - 310 - (1-0-3*1)

1031 = 31*10 - 13*31 + 310 + (1+1)^3

The number with 1031 ones is prime. (i.e. 111.....111 (1031 times) is prime.)

1031 is the first prime after 1024 (2^10 or 32^2 or 4^5).

1031 and 1301 are prime.

1031^2 = 1062961.

1301^2 = 1692601 (reversed of 1031^2).

1031 is the 173rd prime. 173 is also prime.

1031 written in base 2 and base 5 both end in 111.

1031 = 2*3*5 + 7*11*13

304th Day of the Year.

304 = 340 - (40-3) + (4-0-3)

304 = 43 + 30*4 + 34 + 40*3 - (4*3) + (3-0-4)

304 = 3*4*34 - 3^4 - 4^3 + 40 + (4-0-3)

304 = (3+0+4)*43 - (3^0-4)

304 = (43-34)*34 + (4^0-3)

304 = 3^4 + 34 + 4^3 + 43 + (3+0+4)*(3*4) + (4^0-3)

304 = 2*2*2*2*19

222219*304 = 67554576 (only uses 4 numbers, all consecutive)

304 = 8^3 - 7^3 + 6^3 - 5^3 + 4^3 - 3^3 + 2^3 - 1^3

There are 304 semiprimes less than 1024 (2^10, 32^2, 4^5).

304 = 41 + 43 + 47 + 53 + 59 + 61 (6 consecutive primes).

304 = 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 (8 consecutive primes).

###
November 1st (305th Day of the Non-Leap Year)

Nov 1st, 11/1.

111 = 11 + (11-1)^(1+1/1)

111 = 11*11 - (11-1)

111 = 11*(11-1) + 1

111 = (1+1+1)*((1+1+1)*(11+1)+1)

111 = (11+1)*(11-(1+1)) + (1+1+1)

111, 111^2, and 111^3 are all palindromes. (What's the next number with the same property?)

111 = 37*3 (373 is also a palindrome).

111^2 + 111 + 1 is prime.

3^3^3^3^3^3^3^3 has 111 distinct values depending on where you put parentheses.

DigitSum(111^1) = 3 = 3^1

DigitSum(111^2) = 9 = 3^2

DigitSum(111^3) = 27 = 3^3

(What's another number with this property? Can you have fourth powers?)

There are 111 primes less than 610.

111 is divisible by the sum of its digits. (What's another repdigit with this property?)

111 = (1/6)*(1+2+3+....+36)

111, said "eleventy-one", is the age when Bilbo Baggins leaves the Shire in J.R.R. Tolkien's "The Fellowship of the Ring".

There are 111 primes that pass on a clock from 1:00 to 12:59

305th Day of the Year

305 = (30+5)*(3+0+5) + (30-5)

305 = 30*5 + 50*3 + 5

305 = (50-3)*(30/5) + 3*5 + (3+0+5)

305 = (30-5)*(3*5) - 50 - (30-5*(5-0-3))

305 = (30-5)*(3+0+5) + 30*5 - (30+3*5)

305 = (50-3)*(3+0+5) - 35 - (5+3^0)^(5-0-3)

305 = 3^5 + 35 + (30 - (3+0))

305 = 5^3 + 50*3 + 30

305 = 3^5 + 5^3 - 35 - (30-(5-0-3))

305^2 contains 3,0,5 in order. (What's another number like this?)

There are 305 primes between 32^3 and 33^3.

There are 305 primes between 46 and 46^2.

(What other numbers have one or both of these properties?)

305^7 - 304^7 is prime. (What's another number with this property?)

The 51st prime plus the 51st composite is 305 (233+72).

305 is composite but 503 is not. (What's the next number with this property?)

305 written in bases 7 and 8 use the same digits (614 and 461, respectively). (Do other numbers share this property? Other consecutive bases?)

###
November 2nd (306th Day of the Non-Leap Year)

Nov 2nd, 11/2.

112 = 121 - (11-2)

112 = 11*12 - (21-1)

112 = 11^2 - (12-(1+2))

112 = (1+1+2)*(12 + (1*1*2)^(1+1+2))

112 = 12 + 11 + 21 + (1*1*2)*(12 + 11*2)

112 = (1+12) + (11*2) + (11+12) + (21+1) + (21+11)

112^4 contains all odd digits (excluding the ones digit, of course). (Do other numbers share the same property?)

112*211 is palindromic. (What others numbers are like this?)

112^32 + 1 is prime. (What's another number with the same property?)

There are 112 ways to place 5 nonattacking bishops on a 4 by 4 board.

There are 112 primes between 9000 and 10000.

112 = 11 + 13 + 17 + 19 + 23 + 29 (six consecutive primes).

A British long hundredweight is 112 pounds.

112 written in base 36 is less than 36 (34, actually).

112 contains a letter when written in base 11 (the first base that may have letters).

306th Day of the Year.

306 = 36*((60/3) - 3)/(6/3)

306 = (3+0+6)*((6/3)^(6-3^0) + (6/3))

306 = 36 + 63 + 30*6 + (6-3)^3

306 = 60*3 + (6/3)*63

306 = (30+6)*(3+0+6) - 3*6

306 = (60+3)*3 + 63 + 36 + 3*6

306 = (60-3)*6 - 36

306 = (30-6)*(3+0+6) + 63 + (6-3)^3

306 = 2*3*3*17

306*(2 + 3 + 3 + (1+7)) = 4896 (contains only composite digits).

(Are there other numbers with this property?)

306 = 2*3*3*17

306*23317 = 7135002 (contains only noncomposite digits).

(Do other numbers have this property?)

306^2 + 1 is prime.

306^8 + 1 is prime.

306^16 + 1 is prime.

(What other numbers have one, two, or all three of these properties?)

There are 306 primes less than 45^2.

306 is the largest positive number that makes (18*n)/(18+n) an integer.

306^8 + 307^8 is prime. (What's the next number with this property?)

306 + 1 and 306^2 + 1 are prime. (What's a number with both of these properties?)

307^11 - 306^11 is prime. (What's another number with this property?)

306 written in 7 and 15 have the same numbers (615 and 156, respectively).

###
November 3rd (307th Day of the Non-Leap Year)

Nov 3rd, 11/3.

113 = 131 - 3*(3*(1+1))

113 = 31*(1+1+3) - (31+11)

113 = 13*(1+1+3) + 31 + (1+1)^3 + 3^(1+1)

113 = (13+31)*(1+1+3) - 11^(3-1) + (1+13)

113 = (31-13)*(1+1+3) + (11-(1-13))

113 = 11*3 + 31*1 + 1*13 + (3*(1+1))^(3-1^1)

113^3 = 1442897 (contains two squares concatenated....144 and 289). (Do other numbers have this property?)

113 and 311 are primes. (What's another number with the same property?)

Eliminating any digit in 113 still results in a prime. (What's another number like this?)

Any two digit number generated from 113 is prime. (What other numbers display this property?)

113 is the smallest prime factor of 12345678910111213 (concatenation of first 13 natural numbers).

A113 is the license plate on Andy's mom's car in the movie Toy Story.

307th Day of the Year.

307 = 3*73 + 3*7 + 37 + 30

307 = 30*7 + (3+0+7)^(3-7^0) - 3

307 = (30+7)*(3+0+7) - (70-3) + (7-0-3)

307 = 7^3 - (7-3^0)^(3-7^0)

307 = (30-7)*(3+0+7) + 73 + (7-0-3)

307 = (70-3)*(7-0-3) + 37 + (3-7^0)

307^2 = 94249 (palindrome). (What other numbers satisfy this property?)

307 and 113 (referencing Nov 3, 11/3) are both prime. (What's another date with this property?)

307^8 + 308^8 is prime. (What's another number with this property?)

307 is of the form 17^2 + 17 + 1 (Note: 17 is also prime). (What's the next number with this property?)

6^307 - 307^6 is prime (Remark: Only one other known number (>307) satisfies this property). (What is that number?)

307 = 2 + 5 + 41 + 67 + 89 + 103 (primes contains all digits 0-9).

El Salvador has 307 kilometers of coastline.

Excel (and MathCad) cannot assign numbers larger than 10^307.

###
November 4th (308th Day of the Non-Leap Year)

Nov 4th, 11/4.

114 = 141 - 14 - (14-1)

114 = 11*4 + 41*1 + 1*14 + (1+14)

114 = (1+14)*(1+1+4) + 11 - (1-14)

114 = 41 - (1-14)*(1+1+4) - (4+1)

114 = 11*14 - (41-1)

114 = (1+14) + (11+4) + (41+1) + (4+11) + 41 - 14

The sum of the first 114 primes is prime.

There are 114 primes less than 25^2.

114^32 + 1 is prime. (What's the next number with this property?)

3^114 + 10 is prime. (What's the next number with the same property?)

4^4^4^4^4^4^4^4 has 114 different values (depending on where you put the parentheses).

There are 114 primes between 5000 and 6000.

114 written in base 36 is 36. (What other numbers have this property for other bases?)

308th Day of the Year.

308 = 30*8 + 83 - 38 + 3*8 + 3^0

308 = (3+0+8)*38 - 83 - (38-(3+8))

308 = (3+0+8)*(30-8) + 83 - (30-8) + (8-0-3)

308 = (8-0-3)*(80+3) - 83 - 8*3

308 = (30+8)*(8-0-3) + 83 + 8*3 + (8+0+3)

308 = (8-0-3)*(80-3) - (80-3)

308^4 has the run '999'. (Do other numbers have this property? Runs of 4? 5? Other than a '9'?)

308 = 2*2*7*11

308*(2+2+7+11) = 6776 (palindrome).

308*(2 + 2 + 7 + (1+1)) = 4004 (also palindrome).

(Are there others that satisfy this property?)

308 = 2*2*7*11

308*22711 contains all composite digits (22711 is concatenation of prime factors).

(Do other numbers have the same property?)

There are 308 primes of the from x^4 + 1 less than 10^14.

###
November 5th (309th Day of the Non-Leap Year)

Nov 5th, 11/5.

115 = 5*((1+15) + (1+1+5))

115 = 51 + (1+1)^(11-5)

115 = ((1+1)*5)^(1+1) + 15

115 = 11*15 - (51-1)

115 = 11^(1+1) - (11-5)

115 = 15*(1+1+5) + (1+1)*5

115 = 23*5 (23 > 5 but 2 < 3 < 5). (Do other numbers satisfy this property?)

115 = 23*5 (prime factorization only uses prime numbers). (What's the next highest number with this property?)

115*511 = 58765 (contains a run of '8765') (Are there other numbers with the same property? Run of 5? 6? etc.)

115 has 666 pages on the OEIS.

115 + (1+1+5) and 115 + (1*1*5) are one away from a perfect square (11^2). (Do other numbers have this property?)

115^6 + 115^5 + 115^4 + 115^3 + 115^2 + 115 + 1 is prime. (What other numbers are like this?)

5^5^5^5^5^5^5^5 has 115 distinct values (depending on where you put the parentheses).

(The same argument goes for 6,7,8,9, and 10).

115 is composite but its digits are not. (What's another number like this?)

115 = 5! - 5

The oldest verified person is 115 years old. Her name is Misao Okawa. (Curiously, the top ten oldest verified people are all female.)

309th Day of the Year.

309 = 93 + (9-0-3)^(9/3)

309 = 39 + 90*3

309 = 39*(3+0+9) - 93 - (90-3) + 3*9 + (3-0-9)

309 = 93*(9/3) + 30

309 = 39*(9-0-3) + (90-3) - (3+0+9)

309 = 3*9*(3+0+9) - 3*9 + (3+0+9)

There are 309 primes less than 2^11.

4*10^309 + 1 is prime. (Meaning, 4000...000...0001 is prime where there are 308 zeroes.)

If 309 = n, then 2n3n5n7n11n13 is prime (Hence, 23093309530973091130913 is prime). (What's another number with the same property?)

309^2 + 309 + 1 is prime. (What's another number with this property?)

309^32 + 310^32 is prime. (What's the next number with this property?)

309 written in bases 7 and 11 use the same letters (621 and 261 respectively). (What other numbers have a similar property? With different bases?)

###
November 6th (310th Day of the Non-Leap Year)

Nov 6th, 11/6.

116 = 61 + (6-1)*11

116 = 66 + (11-6)*(6-1)*(1+1)

116 = (61+1) + (1+16) + (11+6) + (16+(6-1-1))

116 = 11*16 - (16-1)*(6-1-1)

116 = 11^(1+1) - (6-1*1)

116 = (6-1-1)*(11+16+(1+1))

116^2 = 13456 (1 < 3 < 4 < 5 < 6 and '3456' is a run of 4 consecutive numbers). (Does this happen with other numbers?)

116^2 + 1 is prime. (What's the next number like this?)

116! + 1 is prime. (What's another number with the same property?)

90*116 + 37 and 90*116 + 73 are prime. (What's the next number with both of these properties?)

116^16 + 117^16 is prime. (What's the next number with the same property?)

There are 116 different possible rows/columns in an 11 by 11 crossword puzzle.

The number 1 appears 116 times in the first 1000 digits of pi.

3^116 + 2^115 is prime. (What's the next number with this property?)

116 written as a Roman numeral uses all different letters (CVXI).

The Hundred Years' War between England and France actually lasted 116 years (1337 - 1453).

116 is the record number of wins in Major League Baseball (set by Chicago Cubs in 1906 then Seattle Mariners in 2001).

310th Day of the Year.

310 = 31*10

310 = (10/(3-1-0))*31*(3-1)

310 = 31 + 10 + 30 + 13 + (3+1+0) + (30/(3-1-0))^(3-1-0) - 3

310 = 103 + 130 + 31 + 30 + 13 + 10 + (3-10)

310 = 130 + 10*(10+3+(10/(3-1-0)))

310 = 301 + 3^(3-1-0)

310 = 103*(3+1+0) - 31 - 13 - 30 - 10 - (3+1+0) - (10+(3+1+0))

310*103 and 310*301 use the same numbers (31930 and 93310, respectively). (What other numbers have this property?)

3^310 + 10 is prime (Note: 3 and 10 concatenated give the number 310). (Do other numbers have a similar property?)

(1!)^2 + (2!)^2 + (3!)^2 + ... + (310!)^2 is prime. (What's the next number with this property?)

310 written in base 6 is 1234.

###
November 7th (311th Day of the Non-Leap Year)

Nov 7th, 11/7.

117 = 71 + (7-1-1)*(7+1+1) + 1

117 = 11 + (11+7) - (1-17) + (71+1)

117 = 11*17 - (7-1-1)*(7*(1+1))

117 = 171 - (7-1*1)*(7+1+1)

117 = (7-1-1)*(1+17) + (7+1+1)*((7-1)/(1+1))

117 = (1+1+7)*11 + (1+17)

117 = (11-7)*(11+7) + (7+1+1)*(7-1-1)

117^2 = 13689 (1 < 3 < 6 < 8 < 9). (Do other numbers larger than 117 have this property?)

117*DigitSum(117) is the first four digit number (for 3-digit numbers, abc). (What's the smallest 2-digit number AB such that AB*DigitSum(AB) is a 3-digit number? 4-digit number?)

117^2 + 117 + 1 is prime. (What's another number with the same property?)

117^4 + 117^3 + 117^2 + 117 + 1 is prime. (What other numbers have this property?)

10^21 + 17 is the first prime after 10^21.

10^23 + 17 is the first prime after 10^23.

117 is a divisor of 999999.

There are 117 primes between 6000 and 7000.

There are 117 primes between 18^3 and 19^3.

The 1992 United States men's Olympic basketball team averaged 117 points per game.

311th Day of the Year.

311 = 131 + 113 + 31 + 3*11 + 3*1*1

311 = 113*(3*1*1) - (31-3*1)

311 = 31*11 - (31-1)

311 = (31+1)*(3^(1+1)) + (31-(1+1)^3)

311 = 131 + 3*11*(3+1+1) + (3*1*1)*(3+1+1)

311 = (3*1*1)*(11-3)*(11+3) - (3+1+1)^(3-1*1)

Every permutation of 311 is a prime (31, 13, 11, 113, 131, 311).

311^8 + 312^8 is prime. (What's another number like this?)

311 is a reflective prime (113 is also prime). (What's the next number satisfying this property?)

The digit sum and digit product of 311 are primes (5 and 3, respectively). (What's another number with this property?)

DigitSum(311^1) = 5^1

DigitSum(311^2) = 5^2

(What's the next number with the same property?)

311 is prime. 3 and 11 are primes (concatenation of primes form a new prime). (What's the next number with this property?)

311 = 101 + 103 + 107 (3 consecutive primes).

311 = 53 + 59 + 61 + 67 + 71 (5 consecutive primes).

311 = 31 + 37 + 41 + 43 + 47 + 53 + 59 (7 consecutive primes).

311 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 (11 consecutive primes).

(Note: the number of primes added are prime...--3,5,7,11)

###
November 8th (312th Day of the Non-Leap Year)

Nov 8th, 11/8.

118 = 81 + (11+8) + 18

118 = (81+1) + (8-1-1)^(1+1)

118 = (11-8)*(11+8) + (81-1) - (1+18)

118 = 11*8 + 11 + (1+18)

118 = 11*18 + (11-8)^(8/(1+1)) + 1

118 = 181 - (8+1)*(8-1)

118^2 = 13924 (contains 3,2,4 in order-- 324 = 18^2).

118^4 = 193877776 (run of four 7s). (Are there other numbers with this property or a similar property?)

118 = 2*59 (2 < 59 and 2 < 5 < 9). (What's another number satisfying this property?)

118^4 + 118^3 + 118^2 + 118 + 1 is prime (similarly, 117 had the same property). (Are there other pairs of numbers with this property?)

118^4 + 1 = 193877777 is prime.

118^8 + 1 is prime.

118 = 3^5 - 5^3. Also, 5*118 + 3 and 3*118 + 5 is prime.

118^8 + 119^8 is prime.

118!! + 1 is prime.

118!! - 1 is prime.

118!!!! +1 is prime.

(What's another number satisfying one, two, three, ... of these properties? Any satisfying all?)

The number of primes between 118^2 and 119^2 equals the number of primes between 118 and 2*118. (With what other numbers does this happen?)

There are 118 perfect squares less than 24^3.

1 + 2*118 + 3*118^2 + 4*118^3 is prime (Meaning if 118 = n, 1 + 2n + 3n^2 + 4n^3 is prime). (What's the next number like this?)

312th Day of the Year.

312 = 213 + 123 - (1+23)

312 = 123 + (23-2*3)^(3+1-2)

312 = 132 + 123 + 3*(23-(3+1-2))

312 = 231 + (2^3+1)^(3+1-2)

312 = 31*12 - 31*2 + (3+1-2)

312 = 3*12 + 31*2 + 2*13 + 21*3 + (3+1*2)^3

312 = (3+12) + (31+2) + (2+13) + (21+3) + (3+12)^(3+1-2)

312 = 32*1 + (32+1) + 23*1 + (23+1) + 123 + (2^3-1)*(13-2)

312^4 uses all numbers 3 to 9. (Do other numbers have this or a similar property?)

312 = 2*2*2*3*13 (only uses 1,2,3 in prime factorization). (What's another number like this?)

312^4 + 1 is prime. (What's the next number with the same property?)

312 = 0!*5! + 1!*4! + 2!*3! + 3!*2! + 4!*1! + 5!*0!

1 + 2*312 + 3*312^2 + 4*312^3 is prime (Note: If 312 = n, 1 + 2n + 3n^2 + 4n^3 is prime). (What's the next number with this property?)

There are 312 ways to place 7 nonattacking queens on a 7 by 8 board.

312 written in base 5 is 2222. (What's another number with a similar property?)

312 written in base 7 and 11 have the same digits (624 and 264, respectively). (What other numbers have similar properties?)

312 written in bases 18, 19, 20, ... 36 use at least one letter (except one of the bases). (Which base uses only numbers?)

###
November 9th (313th Day of the Non-Leap Year)

Nov 9th, 11/9.

119 = 91 + 19 +1*1*9

119 = 11*9 + (11+9)

119 = (9+1+1)^(11-9) - (1+1)

119 = (11+9) + (1+19) + (9+11) + (91+1) - (11+19) - (1+1+9) + (9-1*1)

119 = 9^(1+1) + (1+1)*19

119 = 11*19 - 9^(1+1) - 9

119 = (1+1)^(9-1-1) - 1*1*9

119 = (9-1-1)*(9+1+1) + (1+1)*19 + (1+1)^(1+1)

119 = (9-1-1)*(9*1*1) + (11+19) + (19+(9-1-1))

119 = ((9-1-1)-(11-9))! - 1

119^2 + 119 + 1 is prime. (What's the next number displaying the same property?)

1+1+9 = 11 (contained in 119). (What's another number like this?)

There are 119 primes between 4000 and 5000.

2^119 - 119^2 is prime. (What's another number displaying this property?)

10^41 - 119 is the last prime before 10^41.

119^4 + 120^4 is prime. (What's another number with this property?)

The sum of proper divisors of 119 is 25 (1+7+17).

The sum of the divisors of 119 is 144 (1+7+17+119).

(Note: Both are perfect squares).

(Does this happen for other numbers?)

119 is the smallest composite that is 1 less than a factorial (5! - 1).

119 = 17 + 19 + 23 + 29 + 31 (5 consecutive primes).

119 written in bases 2, 5 and 16 are palindromes (1110111, 434, and 77, respectively).

313th Day of the Year.

313 = 133 + 3*(31*(3-1)) - (3*1+3)

313 = (31-13)^(3-1) - (13+(1-3))

313 = (31+13)*(3+1+3) + (3-1+3)

313 = (31+3)*(3*1*3) + (3+1+3)

313 = (31*(13-3)) + 3

313 = (13-1)^(3-1) + 13^(3-1)

313 = (3+13)^(3-1) + 31 + (31-(3-1+3))

313 is a prime of the form n^2 + (n+1)^2 (Here, n = 12). (What's another prime of this form?)

313^16 + 314^16 is prime. (What's the next number with the same property?)

313^2 + 4 is prime. (What's the next number with this property?)

313 is prime. 13 is prime. 3 is prime.

313 is prime. 31 is prime. 3 is prime.

(What other numbers have one or both of these properties?)

You need 313 people to ensure a 50% chance that 5 people share the same birthday.

313 and 119 both have a digit product of 9. (Does this happen with other dates?)

313 is the smallest palindromic prime with only one 1 and with two 3s.

313 written in bases 2 and 13 (and 10) are palindromic (100111001 and 1B1, respectively).

###
November 10th (314th Day of the Non-Leap Year)

Nov 10th, 11/10.

1110 = (10-1*1)*111

1110 = (10-1-1)*(11*10) - 11*11 - 110 + 1

1110 = 101*11 - 1

1110 = (11+10)*(10-1-1)*(1+1+1+0) + 101*((11+1)/(1+1))

1110 = 11^(1+1+1+0) - 111 - 110

1110 = 2*3*5*37 (contains only prime digits). (What other numbers have this property?)

1110 = 2*3*5*37 (Note: 2+35 = 37).

Saying "1110" aloud (meaning, 'one one one zero') describes the number 10 (there is one 1 and one 0). (This actually works for 11/10 through 11/19. It describes the day of the month.)

1110^27 + 1110^25 + 1110^23 + 1110^21 + 1110^19 + 1110^17 + 1110^15 + 1110^13 + 1110^11 + 1110^9 + 1110^7 + 1110^5 + 1110^3 + 1110 + 1 is prime. (What's the next number satisfying this property?)

1110 written in base 5 uses one of each possible digit (13420). (What's another number where this happens?)

1110 written in base 12 uses 6,7,8 (786).

1110 written in base 13 uses 5,6,7 (675).

1110 written in base 16 uses 4,5,6 (456).

314th Day of the Year.

314 = 143 + 134 + (31+(3-1+4))

314 = 431 - 134 + (3+14)

314 = 341 - (13-4)*3

314 = 3*14 + 31*4 + 4*13 + 41*3 + (1-4)^3

314 = 413 - (3+1+4)*(14-3) - (14-3)

314 = 34*(4-1)^(3-1) + (3+1+4)

314 = (3*1*4)*(31-4) - (41-31)

314 = 431 - 143 + (31-4) + 1

Today is known as the second Pi Day (the first Pi Day being March 14th--also known as 3/14).

314^2 + 1 is prime. (What's the next number with this property?)

There are only 4 primes between 31400 and 31499. (Can you name them?)

There are 314 primes less than 10000 with at least one 8 (Note: 3+1+4 = 8).

314^2 + 314 + 1 is prime. (What's another number like this?)

###
November 11th (315th Day of the Non-Leap Year)

Nov 11th, 11/11.

1111 = ((11-1*1)^(1+1) + 1)*11

1111 = 11^(1+1+1) - 11^(1+1) - (11-(1+1))*11

1111 = 111*11 - 11^(1+1) + 11

1111 = (1+1+1+1)*(111+11^(1+1)) + (11+11)*(1+1)^(1+1+1) + 11 - (1+1+1+1)

1111 = (11+11)^(1+1) + (1+1+1+1)*111 + 11^(1+1) + (11+1+1)*(1+1+1+1) + (11-1)

1111 = (11+11)*11*(1+1+1+1) + 11^(1+1) + 11*(1+1)

1111 = 111*(1+1+1)*(1+1+1+1) - 11^(1+1) - (11-1)^(1+1)

1111 = 111*(11-1) + 1

1111 = (11+1)^(1+1+1) - (1+1+1+1)*111 - 11^(1+1) - (1+1+1+1)*(11+1+1)

1111 = (1+1+1+1+11+11)^(1+1) + (1+1+1+1)*111 - (11-(1+1))

1111^2 = 1234321.

1111 = 10^3 + 10^2 + 10^1 + 10^0

DigitProduct(1111^2) = 144 = 12^2.

DigitSum(1111^2) = 16 = 4^2.

(Both are squares.)

(What's the next number with the same property?)

There are 1111 2x2 matrices with entries {1,...30} with a determinant of 1.

The time 11:11 is seen as good luck and many people will wish upon this time.

1111 written in bases 12, 13, and 14 are all palindromes (787, 676, 595, respectively). (What other numbers have a similar property?)

315th Day of the Year.

315 = 31*5 + 3*15 + (13+(3-1)*5)*5

315 = 51*3 + 5*13 + 53 + (3+15)

315 = 153 + (15-3) + 3*1*5 + 135

315 = 351 - (35+1)

315 = 531 - 351 + 135

315 = 35*(3+1+5)

315 = 5!*(5-3+1) - 3*15

315 = 5^3 + 3^(5-1) + 135 - (31-5)

315 = (3*1*5)*(3-1+5)*(5-3+1)

315*513 = 161595 (concatenation of two palindromes). (What's another number with this property?)

315 = 3*3*5*7 (3 â‰¤ 3 â‰¤ 5 â‰¤ 7)

315 = 3*3*5*7

315*(3+3+5+7) begins with '567' (three consecutive numbers).

315 is divisible by its digit sum and digit product. (What's the next number with this property?)

There are 315 primes between 47 and 47^2 (inclusive).

###
November 12th (316th Day of the Non-Leap Year)

Nov 12th, 11/12.

1112 = 1121 - (1+1+1)^2

1112 = 1211 - (1+1+1+2)*(21-1) + 1

1112 = 111*2 + 112 + 121 + 211 + 111 + 21*11 + (2^(1+1+1))*(11+2)

1112 = ((21-(2+1)) + 11^2)*(2^(1+1+1))

1112 = (11+12+21)*((2+1)^(1+1+1)) - (1+1+1+2)*(11+1+2) - (1+1+1)*2

1112 = ((1+1+1)*2)*21*11 - 211 - 2^(2*(1+1+1)) + 1

1112*2111 is a palindrome. (What's another number like this?)

DigitProduct(1112) = 2 (prime).

DigitSum(1112) = 5 (prime).

(What's the next number with the same property?)

1112^5 reversed is a prime (Note: 1+1+1+2 = 5). (What other numbers satisfy this property?)

1112 written in base 3 begins with '1112'. (Are there other numbers with this property? In another base?)

316th Day of the Year.

316 = 136 + 163 + (3*6-1)

316 = 361 - 3*16

316 = 3*16 + 31*6 + 6*13 + 61*3 + (163+16)

316 = (631 + (6/3-1))/(6-3-1)

316 = 613 - (3*1*6)*16 - (3+6)

316 = (3+1+6)^(6-1-3) + 6^(3*1)

316 = (31+6) + (3+16) + (6+13) + (61+3) + 163 + (6+1)*(3-1)

316 = (16+63)*(6+1-3)

316 = (6^(3-1))*(3+6-1) + (6+1)*(3+1)

316 = 2*2*79

2+2+79 = 83 (prime factors added is a prime). (What's another number with the same property?)

There are 316 primes of the form x^2 + 1 less than 10^7.

316^23 + 316^21 + 316^19 + 316^17 + 316^15 + 316^13 + 316^11 + 316^9 + 316^7 + 316^5 + 316^3 + 316^1 + 1 is prime. (What's another number with this property?)

316^8 + 317^8 is prime. (What's the next number with this property?)

316 is the largest number such that its square has 5 digits.

316 written in bases 16 through 34 include at least one number. (Do other numbers have bigger gaps?)

316 written in base 7 is 631 (uses 3,1,6). (Do other numbers have this property? With other bases?)

###
November 13th (317th Day of the Non-Leap Year)

Nov 13th, 11/13.

1113 = 311 + 113 + 131 + 3*111 + (13+1+1)^(1-1-1+3)

1113 = 1131 - 3*(1+1+1+3)

1113 = 1311 - (131+(1+3+1)) - 31*(1+1)

1113 = (11+13)*(31+11) + 113 - (1+1+1)^3

1113 = ((1+1)*31)*(1+1+1+3)*(1*1*1*3) - 3

1113 = (31+1+1)^(1+1) + 3*(1+1+1)^3

1113^2 = 1238769 (uses two strings of consecutive numbers, '123' and '6789'). (Do other numbers have this property? With different strings?)

1113 contains three 1s and one 3 (Note the numbers used in that sentence).

2^1113, 3^1113, 5^1113, 7^1113, 11^1113, and 13^1113 have even digit sums (Note: 2,3,5,7,11, and 13 are prime). (What's another number that displays this property?)

1113^47 + 1113^45 + 1113^43 + 1113^41 + ... + 1113^5 + 1113^3 + 1113^1 + 1113 is prime. (What's the next number where this property holds?)

1113 = 2^3 + 3^4 + 4^5

1113 written in base 13 and 12 is 678 and 789, respectively (consecutive numbers). (Is this similar to other numbers in other bases?)

1113 = 3*7*53 (only uses prime digits in prime factors). (What other numbers have this property?)

The sum of the divisors of 1113 is 1728 (a perfect cube--12^3). (What's another number with the same property?)

317th Day of the Year.

317 = 371 - (7+3-1)*(13-7)

317 = 137 + 173 + 7

317 = 137 + (3+7)*(3*1*7-(7-3-1))

317 = 173 + (3*1*7-(3+7-1))^((7-1)/3)

317 = 713 - 371 - (7-(3-1))^((17-3)/7)

317 = 31*7 - 3*17 - 7^(3-1)

317 = (7+1)*13 + 71*3

317 = 37*1 + 73*1 + (17-3)^(3-1) + (3+1+7)

317 = (3*1*7)^(3-1) - (3+1+7)^((7-1)/3) - (7-3-1)

317 = (-3)^3 + 1^3 + 7^3

317 has 317 pages on the OEIS as of Nov 13, 2013. (Is there another number like this?)

(10^317 - 1)/9 is prime (Meaning, 111...111 with 317 ones is prime). (What's another number satisfying this property?)

317 is the concatenation of 3 and 17 (two primes) or 31 and 7 (also two primes).

317 is prime. (Are there other numbers like this?)

There are 317 primes less than 2100.

Removing any digit in 317 keeps the number prime. (What's another number like this?)

2*3*5*7*11*13*17*...*317 - 1 is prime. (What's another number with this property?)

317 is prime.

317^2 + 4 is prime.

(Any other primes with this property?)

###
November 14th (318th Day of the Non-Leap Year)

Nov 14th, 11/14.

1114 = 411 + 114 + 141 + 4^(1+1+1)*(4+1+1+1)

1114 = 4*111 + 41*11 + 4*11 + 14 + 11*14 + (1+1+1+4)

1114 = 114*11 - 14*(11-1)

1114 = 111*14 - 111*4 + 1*1*1*4

1114 = (1+1+1+4)*141 + 114 + (14-1*1)

1114 = 141*11 - 4*111 + (1+1+1+4)

1114 = 1141 - (1*1*1*4)*(1+1+1+4) + 1

1114 = 1411 - 141 - 114 - (41+1)

1114 = 2*557 (only uses prime digits).

1114 = 1^2 + 2^3 + 3^4 + 4^5

1*1*1*4 = 4 (1114 has 4 divisors). (What's another number like this?)

1114 is the smallest 4-digit number with a digit product of 4.

318th Day of the Year.

318 = 138 + 183 - 3

318 = 381 - 3*(3+18)

318 = 31*8 + 3*18 + (3-1)*8

318 = 8*13 + 81*3 - 3*1*8 - (8-3*1)

318 = 1*83 + 38*1 + 183 + (3-1)*(8-1)

318 = (3+1+8)*(3*1*8) + (3+18) + (8+1-3)

318 = 138 + 18*(8-1+3)

318 = 183 + (8-1*3)*(3*(1+8))

318 = 31*18 - (8-1+3)*(3*1*8)

318 = 2*3*53 (all prime digits--just like 1114). (What other dates have this property--both day and date satisfying it?)

318*(3*1*8) = 7632 (7 > 6 > 3 > 2). (Do other numbers have this property?)

318^16 + 1 is prime. (What's another number with this property?)

If 318 = n, 2n3n5n7n11 is prime (Meaning, 231833185318731811 is prime). (What's the next number with this property?)

According to the The Simpsons, 318 is the Police code for waking an officer.

The mass of Jupiter is approximately 318 times the Earth.

###
November 15th (319th Day of the Non-Leap Year)

Nov 15th, 11/15.

1115 = 511 + 111*5 + (5+1+1)^(5-1-1-1)

1115 = 5*(111*(5-1-1-1)) + 1*1*1*5

1115 = 15*(15*(5*1*1*1)) - (5-1-1-1)*5

1115 = 11*(111-(1+1+1+5)) + (5+1)*(1+1+1)

1115 = 5^(1+1+1) + (1+1+1)^5 + 51*11 + 151 + 5*(5+1+1)

1115 = 111*15 - 5*111 + 1*1*1*5

1115 = 11*115 - 151 + 1

1115 = (11+15)*(51-11) + 15*(1*1*1*5)

1115 = (51+11)*(1+1+15) + (51+1+1) + (5+1+1+1)

1115 = 1151 - (5+1)^(1+1)

1115 = 1511 - 51*11 + 151 + (15-1*1)

1115*1115 uses 1,2,3,4,5.

1115*5111 uses 5,6,7,8,9.

(Is there another number with this property or a similar property?)

1115 = 5*223 (only uses prime digits).

1115 = 5*223 (223 > 5 but 2 â‰¤ 2 â‰¤ 3 â‰¤ 5).

If 1115 = n, then 2n3n5n7 is prime (Meaning, 2111531115511157 is prime). (What's another number with this property?)

1115 is the smallest four digit number that produces a prime when divided by the product of its digits.

1115 = 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 (nine consecutive primes).

319th Day of the Year.

319 = 31*9 + 3*19 - (13 + (9/3+1))

319 = 91*3 + 9*13 - 93 + (3+19)

319 = 93 + 39 + 193 + (3*1-9)

319 = 31*19 - (3*1*9)*(13-9/3)

319 = (31+9) + (3+19) + (91+3) + (9+13) + 139 + (9/3-1)

319 = 193 + 139 - 13

319 = 391 - (9*(3-1+9) - (3*1*9))

319 = (3+1+9)*(3+19) + (9/3*1)*(3-1+9)

319 = (3*1*9)*(3*1+9) - (9-3-1)

319 = (3-1+9)*(3*1*9+(9/3-1))

319^3 = 32461759 (all different digits). (This is the largest number with this property. What's the second to last number with this property?)

319 = 11*29 (11 < 29 and 1 â‰¤ 1 â‰¤ 2 â‰¤ 9).

10^29 + 319 is the first prime after 10^29.

319^10 + 319^9 + 319^8 + 319^7 + 319^6 + 319^5 + 319^4 + 319^3 + 319^2 + 319^1 + 1 is prime. (What's another number with this property?)

There are 319 primes less than 46^2.

319 is a song by Prince.

319 = 103 + 107 + 109 (three consecutive primes). (Note: these primes were used above for 1115.)

319 contains 6 consecutive 1s in binary.

###
November 16th (320th Day of the Non-Leap Year)

Nov 16th, 11/16.

1116 = 611 + 111*6 - 161

1116 = 161 + 116 + 611 + 11*16 + 61 - (1+1+1+6)

1116 = 61*11 + 611 - 116 - (61-11)

1116 = 111*16 - 611 - (6+1)^(1+1)

1116 = 11*116 - 116 - 11*(6-1-1*1)

1116 = (1+1+1+6)*(116 + 16/(1+1))

1116 = (1*1*1*6)*(161 + (6-1)^(1+1))

1116 = 111*(1+1+1+6) + 161 - 11*(6*(1-1/(1+1+1)))

1116 = 61*(1+1+16) + (1+1+1)*6

1116 = (11+1+6)*(61+1)

1116 = 1161 - (1+1+1+6)*(6-1)

1116 = (61+11)*(16-1) + 6^(6/(1+1+1))

1116 = (61-11)*(16+1*1*1*6) + 16

1116 = (11+16)*((1+1+1)*6 + (16+1*1*1*6)) + (6/(1+1+1))*(11+1+6)

1116 = (1+1+1)^6 + 6^(1+1+1) + 116 + (61-6*1)

1116 = 11^(1+1+1) + 1^(1+1+1) - 6^(1+1+1)

1116 = (11+16)^(6/(1+1+1)) + 11*16 + 161 + (61-11)

1116 = (6/(1+1+1))^(6+1+1+1) + 611 - (6+1*1*1)

1116 is divisible by its digit product (6*186) and its digit sum (9*124). (What's another number with this property?)

1116^13 + 1116^11 + 1116^9 + 1116^7 + 1116^5 + 1116^3 + 1116 + 1 is prime. (What's the next number with this property?)

1116^2 + 1116 + 1 is prime. (What's another number with the same property?)

320th Day of the Year.

320 = 203 + 23*(3+2+0) + 2

320 = 230 + 3*20 + 30

320 = 3*20 + 32 + 23 + 30*2 + 230 - (3+2+0)*(30/2) - (3^2+2^0)

320 = (3^2)*32 + (30+2)

320 = (2^3)*(32+2^3)

320 = 23*(30/2) - (3+2)^2

320 = (20-3)^2 + (32-(3-2))

320 = (30-2)*(3*2+(3+2)) + (20-3) - (3+2+0)

320 = (3+2+0)*(2^(3*2))

Every nonzero digit and every sum of any number of digits of 320 is prime (i.e. 3, 2, 3+2, 3+2+0, 3+0, 2+0, ...etc are all prime).

320! + 1 is prime. (What's the next number with the same property?)

320^4 + 1 is prime. (What's another number like this?)

320^6 + 320^5 + 320^4 + 320^3 + 320^2 + 320 + 1 is prime. (What's the next number satisfying this property?)

There are 320 possible moves a queen could make on a 5 by 5 chessboard.

There are 320 primes between 33^3 and 34^3.

320 is the largest determinant that can be obtained from a 10 by 10 matrix of zeroes and ones.

###
November 17th (321st Day of the Non-Leap Year)

Nov 17th, 11/17.

1117 = 711 + 171 + 117 + (71+11) + (7-1)^(1+1)

1117 = 111*7 + 171 + (7-1-1-1)*(17-(7-1-1-1)) + 117

1117 = 111*17 - 111*7 + 1*1*1*7

1117 = (7-1-1-1)*(117+171) - 7*(7-1-1*1)

1117 = 7*(111+(7+1)*(7-1)) + (7-1-1-1)

1117 = (11+17)*((1+1+1)*(17-(7-1-1-1))) + (7-1-1)^(1+1)

1117 = (17-11)*(171+11) + (17+(1+7))

1117 = (71+11)*(17-(7-1-1-1)) +17*(1+1+1)

1117 = 11*117 - 17*(1+1+1+7)

1117 = (11*17)*(17-11) - (7-1-1)

1117 = 1*117 + 11*17 + 111*7 + (7-1)^(1+1)

1117 = 71*(17-1-1) + (7-1-1-1)*(17-(7-1-1-1))

1117 = 1171 - (17-11)*(7+1+1)

1117 = 1711 - 117 - 171 - 11*17 - 71 - (7-1)*(7+1)

Saying "1117" digit-by-digit (i.e. "one one one seven") describes the number 17.

1117 is the smallest prime with 3 consecutive identical digits.

There are 1117 primes less than 9000.

1117^2 has all different digits.

1117 is prime.

Inserting a digit between adjacent digits of 1117 is not prime (Meaning, 1010107, 1111117, 1212127, etc. are composite). (What's the next prime number satisfying this property?)

If 1117 = p, p3p and 3p3 are primes (111731117 and 311173, respectively).

1117 or "one thousand one hundred seventeen" uses 5 words (5 is prime along with 1117). (What's another prime like this?)

1117 is the 187th prime (Note: 187 = 11*17).

1117^2 + 4 is prime. (What's another prime number with this property?)

The 1117th prime is 8999 (has the same property of 1117--3 of the same digit and a fourth different one).

1117 is the smallest four-digit prime comprised of 2 two-digit primes (11 and 17).

Any two digits of 1117 will result in a prime.

Inserting a single 7 between any two digits of 1117 is still a prime (i.e. 11177, 11717, and 17117 are prime).

321st Day of the Year.

321 = 123 + 231 - 3*(32-21)

321 = 213 + 123 - (12+3)

321 = 132 + 213 + (3+2-1)*(3*2*1)

321 = 231 + 132 - (2^3-1)*(3+2+1)

321 = 32*21 - 231 - 123 + (3*(2-1))

321 = 12*23 + (3+2)*(3^2)

321 = 3*21 + 32*1 + 12*3 + 1*23 + 132 + (2^3-1)*(3+2)

321 = (32+21)*(3+2+1) + (3*(2-1))

321 = (32+1) + (3+21) + (12+3) + (1+23) + (12+3)^(3-2+1)

321 = 21*(12+3) + (3*2*1)

321 = (32+1) + (12+3) + (31+2) + (13+2) + (1+23) + (3+21) + 132 + (3^2)*(3+2)

321 = 13*(3+21) + 3^2

321 = (21-3)^(3-2+1) - 3*(2-1)

321 = (13-2)*(31-2) + (3-2+1)

321 = (12-3)*((3+2+1)^(3-2+1)) + 3*(1-2)

321 = (21/3)*((3^2)*(3+2)) + (3*2*1)

321^16 + 322^16 is prime. (What's the next number with this property?)

321^19 + 321^17 + 321^15 + 321^13 + 321^11 + 321^9 + 321^7 + 321^5 + 321^3 + 321 + 1 is prime. (What other numbers have this property?)

The digit 8 appears 321 times in all the four-digit primes.

The sum of divisors of 321 is 432 (similar format as 321).

###
November 18th (322nd Day of the Non-Leap Year)

Nov 18th, 11/18.

1118 = 811 + 181 + 118 + 1*1*1*8

1118 = 111*8 + 181 + (8-1)^(1+1)

1118 = 111*18 - 111*8 + 1*1*1*8

1118 = 81*11 + 11*18 + (11+18)

1118 = 11*118 - ((1+1)*18)*(8-1-1-1)

1118 = 1*118 + 11*18 + 111*8 - 81 - (8-1-1-1)

1118 = (1+1)^(1+8) + 8^(1+1+1) + 81 + (18-(8-1-1-1))

1118 = 18*((8-1)*(8+1)) - (18-1-1)

1118 = (11+18)*((1+1)*(1+8) + (1+1+18)) + (1+1)*(1*8)

1118 = (11*18)*(8-1-1-1) + (1+1)^(8-1)

1118 = 81*((8-1)*(1+1)) - (18-1-1)

1118 = 11*(111-8) - (8-1-1-1)*(1+1+1)

1118 = (111+8)*(1+8) + (1+1+1)*8 + (18+(8-1-1-1))

If 1118 = n, 2n3n5n7 is prime (Meaning, 2111831118511187 is prime). (What's another number with this property?)

1118 is composite but 8111 is prime. (What other numbers satisfy this property?)

1118 written in bases 11 and 12 use the same digits (927 and 792, respectively). (Are there others that have a similar property? Different bases?)

322nd Day of the Year.

322 = 223 + 3*(22+(3^2+2))

322 = 232 + (2+3+2)*(2*(3+2) + 3*(2/2)) + (3-2-2)

322 = 3*22 + 32*2 + 2*23 + 22*3 + ((3+2)*2)*(3*2+2)

322 = (32+2) + (3+22) + (2+23) + (22+3) + 3*(3^(2+2) - (3+2)*2)

322 = (3*2*2)*(3+22) + (32-2*(3+2))

322 = (3+2+2)*(2*23)

322 = 2^(2^3) + 22*3

322 = (3^(2+2))*((2^3)/2) - (3-2/2)

322 = 2*7*23 (uses only prime digits and uses 3,2,2 in prime factorization). (Do other numbers have both of these properties?)

322 is composite but both 233 and 223 are prime. (What other numbers have the same property?)

322 is the 12th Lucas number (Note: 12 = 3*2*2). (Do other Lucas numbers share this property?)

322 is the smallest number whose square has 6 distinct digits (103684).

There are 322 different ways to rearrange 1,2,3,4,5,6,7 so no consecutive numbers are next to each other.

323^11 - 322^11 is prime. (What's the next number with this property?)

322^2 + 323^2 + 324^2 is prime. (What's another number like this?)

There are 322 composites less than 20^2 (including 20^2).

322 is a composite number consisting of all prime digits. (What's a composite number with this property that doesn't use the number 2?)

If 322 = n, 2n3n5n7n11 is prime (Meaning, 232233225322732211 is prime). (What's the next number with the same property?)

###
November 19th (323rd Day of the Non-Leap Year)

Nov 19th, 11/19.

1119 = 911 + 191 + (19-1-1)

1119 = 111*9 + (111+9)

1119 = 911 + 119 + (91-1-1)

1119 = 91*11 + 11*19 - 91

1119 = 1*119 + 11*19 + 111*9 - 191 - (19-1-1)

1119 = 91*11 - (1-119)

1119 = 91*(1+1+1+9) + (1+1+1)*9

1119 = (91+11)*(9+1+1) - (1+1+1)

1119 = (91+1+1)*(1+1+1+9) + 9/(1+1+1)

1119 = 19*((1+1)*(11+19)) - (1+1+19)

1119 = (1+1+19)*((9/(1+1+1))*(1+1+1+9) + (19-1-1)) + (9-1-1-1)

1119 = (11+19)*((9-1-1-1)^(1+1)) + ((9/(1+1+1))*(1+1+1+9) + (1+1+1))

1119 = (1*1*1*9)*(119) + (((9-1)/(1+1))*(1+1+1+9))

1119 = 9^(1+1+1) + (1+1)^(1*9) - 119 - (9/(1+1+1))

1119^2 = 1252161 (concatenation of 3 cubes: 125//216//1). (Are there other numbers with this property?)

1119 = 3*373 (only uses prime numbers in prime factorization).

1119^19 + 1119 + 1 is prime (Note the exponent '19' is in '1119'). (What's another number with the same property?)

323rd Day of the Year.

323 = 233 + (2+3+3)*(2^3+3) - (3-2-3)

323 = 332 - (3*2+3)

323 = (3*2*3)^(3+2-3) - (2-3/3)

323 = 3*23 + 32*3 + 33*2 + (23*(3-2+3))

323 = (3+23) + (32+3) + (33+2) + ((3+2)*3)^(3+2-3) + 2*(3/3)

323 = 23*(23-(3+2+3)) - (23+(2-3))

323 = 32*(3*2*3 - (3+2+3)) + 3^(3-2)

323 = (3+23)*(3^2+3) + (3+2^3)

323 = (32+3)*(3*2+3) + (3+2+3)

323 = 17*19

(1+7) + (1+9) = 18 (18 = 3*2*3). (Does this happen with other numbers?)

Inserting a 9 between adjacent digits in 323 results in a prime (i.e. 3923 and 3293 are prime). (What's the next number with this property?)

323 is the smallest natural number that uses 26 letters in its English spelling, including the "and" ("three hundred and twenty three").

There are 323 perfect squares less than 47^3.

323 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 (nine consecutive primes).

23^323 + 2 is prime. (Note this is of the form 23^n + 2. 23 and 2 are both in 323). (What's another number with this property?)

323 is also palindrome if it's written in bases 14 and 18 (191 and HH, respectively). (What other palindromes in base-10 are palindromes in other bases? Three other bases? Four?)

323 is the smallest palindrome that is the product of two twin primes (17*19).

###
November 20th (324th Day of the Non-Leap Year)

Nov 20th, 11/20.

1120 = 1102 + (20-1-1)

1120 = 1201 - (11-2)^(1*1*2)

1120 = 1210 - (11-2)*(20/(1+1))

1120 = 2011 - 11*((11-2)^(1*1*2))

1120 = 11*120 - 2*(10^(1*1*2))

1120 = 11*20 + 21*10 + 112 + 120 + 211 + 11^2 + (11-2)*(12+1*1*2)

1120 = ((1+1+2+0)^(1+1+2+0))*(1+1+2+0) + 11^2 - (10/(1*2))^(1*1*2)

1120 = 11^(1+1*2+0) - 211

1120 = (1+1+2+0)*(211 + (2^(10-(1+1+2+0))) + (10/(2*1)))

1120 = 11*102 - 1*1*2

1120 = (11+20)*((10-(1+1+2+0))^(1*1*2)) + (1+1+2+0)

1120 = (20-11)*((1+120)+(12/(1+1+2+0))) + (1+1+2+0)

1120 = 7*5*2*2*2*2*2

7522222*1120 (10 digits) uses only even digits (Note: 7522222 is the concatenation of prime factors in decreasing order). (Do other numbers have the same property?)

1120^8 + 1 is prime. (What's another number that ends in a 0 with this property?)

1120^64 + 1 is prime (Note: this number ends in 63 zeroes and a 1 in the ones place). (What's the next number with the same property?)

1120 written in bases 3, 11, and 34 (1112111, 929, and WW, respectively).

324th Day of the Year.

324 = 432 - (4+3+2)*((24-3)-(3+2+4))

324 = 243 + (3+2+4)^(2*(4-3))

324 = 234 + (4+2+3)*(4+2*3)

324 = (32/4+(4+2*3))^((4+2)/3)

324 = 3*24 + 32*4 + 4*23 + (2*(4-3))^(4+3-2)

324 = 43*2 + 2*34 + 42*3 + (3+2*4)*((3-2)*4)

324 = (32+4) + (3+24) + (4+23) + (42+3) + (34+2) + (43+2) + (4+2^3)*(3+2+4)

324 = (3+2+4)*((3*(4-2))^(3*2-4))

324 = (3*2*4)*(3^2+4) + (3+2+4)

324 = (3+24)*(3*2+4)

324^3 has digits no higher than 4. (Note: 4 is the highest number in 324. Do other numbers have this property or a similar property? Fourth power?)

324*423 has each non-composite digit (137052). (Are there other numbers like this?)

324 = (2^2)*(3^4) (uses only the digits in 324). (What other numbers satisfy this property?)

324! - 1 is prime. (What's another number with this property?)

There are 324 primes less than 10000 with at least one 5.

The concatenation of 324^2, 324, and 1 is prime (that is, 1049763241 is prime). (What's the next number with this property?)

324 = 73 + 79 + 83 + 89 (four consecutive primes).

Besides base-10, 324 is also a square if written in base 9, 15, 16, 17, and 18 (400, 169, 144, 121, 100). (Is there a number that is a square is more than 6 bases?)

324 is 169 base 15 (13^2).

324 is 144 base 16 (12^2).

324 is 121 base 17 (11^2).

324 is 100 base 18 (10^2).

(Do other numbers have this property?)

###
November 21st (325th Day of the Non-Leap Year)

Nov 21, 11/21.

1121 = 11*121 - (211-1)

1121 = 1112 + (11-2*1)

1121 = 11*21 + 112 + 121 + 211 + (1+1+2+1)*(111-21) - (1+1)^(2*1)

1121 = (1+1+2+1)*211 + 11*((1+1+1)*2)

1121 = 21*((11+21)+(12+11)) - (21-(2+1+1))*(1*1*2*1)

1121 = (11+21)*((1+1+2+1)*(21/(1+1+1))) - (12-11)

1121 = (12+11)*(12*(1+1+2)) + (12+(1+1+2+1))

1121 = (11*(2+1))^(1+2) + (11+21)

1121 = (21-1-1)*((2+1)*(21-1-1) + 1*1*2*1)

1121*1211 = 1357531 (palindrome where 1 < 3 < 5 < 7 > 5 > 3 > 1). (Are there others with a similar property? 2468642?)

1121*21 and 1121*12 use 1,2,3,4,5 only once (23541 and 13452, respectively). (Do other numbers have this property?)

If 1121 = n, 2n3n5n7 is prime (Meaning, 2112131121511217 is prime). (What's the next number in the sequence?)

1121 written in base 2 has '1121' in it. (What's another number like this?)

(1121! - 7)/7 is prime. (Only one other known number higher than 1121 satisfies this. What number is that?)

If 1121 = n, 987654321n is prime (Meaning, 9876543211121 is prime). (What's another number that satisfies this property?)

325th Day of the Year.

325 = 235 + (2*5)*((5-2)*3)

325 = 253 + (5-(3-2))*(23-5)

325 = 352 - 3^(5-2)

325 = (32+5) + (3+25) + 253 + (2*5-3)

325 = (5+23) + (52+3) + 253 - (3*2+5)

325 = (53+2) + (35+2) + 235 - (2^3-5)

325 = 3*25 + 32*5 + (2*3*5)*(5-2)

325 = 5*23 + 52*3 + (5-2)*(23-5)

325 = 35*2 + 2*53 + 5^3 + (5-2)*(2^3)

325 = (3*2*5)*(3*2+5) - 25/(3+2)

325 = (3+2+5)*(2^5) + 5^(3-2)

325 = 25*(5*3-2)

325 = (32+5)*(3^2) - 2^3

325 = (3+25)*(3*2+5) + (3*5+2)

325 = ((5-2)*(2*3))^(5-3) + (5-3)/2

3251, 3253, 3257, and 3259 are all prime. (What's the next number with the same property?)

325^19 + 325^17 + 325^15 + 325^13 +325^11 + 325^9 + 325^7 + 325^5 + 325^3 + 325 + 1 is prime. (What's another number with this property?)

12^325 - 325^12 is prime. (What's the next number with this property?)

325 divides (5^325 - 3^325 - 2^325). (Are there other numbers with this property?)

325 written in bases 2, 4, 8, 18, and 24 are palindromes. (What other base-10 non-palindrome has a similar property?)

###
November 22nd (326th Day of the Non-Leap Year)

Nov 22nd, 11/22.

1122 = 112 + 122 + 211 + 221 + 112*2 + 121*2 - (12-1*2)

1122 = 11^(1+1+2/2) - (211-2)

1122 = 12^(1+2) - 22^(1+1) - 122

1122 = 11*122 - (1+2+2)^(2+2-1) - (22-(1+2))*(1+2+2)

1122 = 11*22 + 112*(2^(1+2)) - 2^(1+1+2)

1122 = (11+22)*((22-(2+2+1))*2)

1122 = 2*3*11*17

2+3+11+17 = 33 (1122 is divisible by 33).

326th Day of the Year.

326 = 263 + (3-2+6)*(6*3/2)

326 = 236 + (3+2)*((3^2)*(6/3))

326 = 362 - (3*2)^(6/3)

326 = (32+6) + (3+26) + 263 - (6*2/3)

326 = (6+23) + (62+3) + 236 - ((6/3)+2)

326 = 3*26 + 32*6 + (2^3+6)*(6*2/3)

326 = 6*23 + 62*3 + (2^3-6)

326 = 36*2 + 2*63 + 2^(3-2+6)

326 = (3+2+6)*(3+26) + (3-2+6)

326 = (3*2*6)*(6*3/2) + (2^3-6)

326 = (32+6)*(3*(6/2)) - (6-2)^(6/3)

326 + 3*2*6 = 362

326^2 + 1 is prime. (What's another number with the same property?)

326^16 + 327^16 is prime. (What's the next number with this property?)

Of all the four digit primes, there are 326 4's.

There are 326 primes less than 10000 with at least one 4 (Note: this and the statement above are different).

326 = 251 + 75 (54th prime plus 54th composite)

326 prefixed or followed by any digit still remains composite. (What's another number with this property?)

###
November 23rd (327th Day of the Non-Leap Year)

Nov 23, 11/23.

1123 = 1213 - (1*1*2*3)*(13+1*2)

1123 = 1231 - (3^2)*(23-11)

1123 = 1321 - 11*(11+(1+1+2+3))

1123 = 1132 - 1*1*3^2

1123 = 1312 - (1+2^3)*(32-11)

1123 = 112 + 123 + 231 + 321 + 312 + (1*1+23)

1123 = 213 + 113 + 131 + 311 + 211 + 132 + (23-11)

1123 = 11*123 - (231-1)

1123 = 11*23 + 112*3 + 32*11 + 123 + (11+23) + (2+3)^(1+1)

1123 = (11-2)^3 + 321 + 11*2*3 + (1+1+2+3)

1123 = 12^(1*3) - 3*211 - (1+1+2+3)*(1+1-2*3)

1123 is the concatenation of the first four nonzero terms of the Fibonacci sequences. (1123 is also prime. What's another prime with this property?)

1123 = 33^2 + 33 + 1 = 34^2 - 34 + 1

Saying 1123 digit-by-digit ("one one two three") spells out the number 133.

1123 is prime.

1^1+1^1+2^1+3^1 = 7 is prime.

1^3+1^3+2^3+3^3 = 37 is prime.

1^5+1^5+2^5+3^5 = 277 is prime.

(Do other numbers have all these properties?)

1123 and 2311 are primes and both the concatenation of two two-digit primes. (Are there other days of the year with this property?)

1+1 (first digits), 1+2 (middle digits), 2+3 (last digits), and 1+1+2+3 is prime. (What's another prime like this?)

327th Day of the Year.

327 = 273 + (2+7)*(3*2)

327 = 237 + (3^2)*(7+3)

327 = 372 - (27+3) + 3*(2-7)

327 = (3*2*7)*(3-2+7)

327 = (3+2+7)*27 + (7+2)/3

327 = 32*7 + 3*27 + (2*7-3)*((7-3)/2)

327 = 7*23 + 72*3 - ((7-3)/2)*((7-2)^(3^2-7))

327 = (32+7) + (3+27) + (72+3) + 7+23) + 2^7 + (7-2)^((7-3)/2)

327 = (32+7)*((7-3)*2) - 3*(2-7)

327 = (3+27)*(7*2-3) - (7+2)/3

There are 327 primes less than 2200.

The first prime after 10^52 is 10^52 + 327.

327!! + 32 is prime (Only one known number higher). (What is that number?)

There are 327 primes less than 3^7.

###
November 24th (328th Day of the Non-Leap Year)

Nov 24, 11/24.

1124 = 241 + 412 + 421 + (1+1)*(1+24)

1124 = 411 + 141 + 124 + 121 + 142 + 112 + (1*1*2*4)*(2^(4-1)) + (4-2-1)

1124 = 11*124 - 24*(24-(2^4-1-1))

1124 = 11*24 + 421 + 412 + (4-1)^(2+1)

1124 = 42*11 + 4*211 - 214 + 2^(1+4)

1124 = (1+1+2+4)*(14*(1+1+2*4))

1124 = (11+24)*(2^(1+4)) + (4/2+1+1)

1124 = (42+11)*((4-1)*(4+2+1)) + (1+(1+4)*2)

1124 = (1+1+24)*(42+1) + 24/(1+1+2)

1124 = (11+2+4)*(11*(2+4)) + (4/2)

DigitProd(1124) = DigitSum(1124) (Besides 1,1,2,4, what other digit arrangement has this property?)

There are 1124 ways to place 3 non-attacking bishops on a 5 by 5 board.

There are 1124 semiprimes less than 2^12.

Only 4 primes are between 112400 and 112499. (Can you name one? All four?)

1124 written in bases 13, 16, 19 and 22 are palindromic (Note: the bases are three apart from each other). (What other non-palindrome has this property?)

328th Day of the Year.

328 = 283 + (3^2)*(8-3)

328 = 238 + (2+8)*(8-2+3)

328 = 382 - (8-2)*(8+3-2)

328 = 3*2*8 + 283 - (8-2-3)

328 = 3*28 + 32*8 - 3*(8/2)

328 = 8*23 + 82*3 + (23-(8-2))*(3*2)

328 = 38*2 + 2*83 + (83+(8-3-2))

328 = (3*2*8)*(8/2+3) - (2^3)

328 = (3+2+8)*(28-3) + (8-3-2)

328 = (3+28)*((8-3)*2) + (8-2)*3

328 = (32+8)*((3-2)*8) + (2^3)

328 = (32+8) + (3+28) + (8+23) + (82+3) + ((3*8)/2)^(8-3*2) - (8-3-2)

The sum of the first 15 primes is 328 (2+3+5+7+11+13+17+19+23+29+31+37+41+43+47 = 328).

328^4 + 1 is prime. (What's the next number with the same property?)

The sum of the first 328 primes is prime. (When will this happen next?)

If 328 = n, n0123456789 is prime (Meaning 3280123456789 is prime). (What's the next number with this property?)

###
November 25th (329th Day of the Non-Leap Year)

Nov 25, 11/25.

1125 = 512 + 112 + 511 - 1*1*2*5

1125 = 521 + 115 + 151 + 152 + 125 + (52+(5*2-1*1))

1125 = 11*125 - (11+5)^2 - (1+1)*(2-5)

1125 = 5*211 + (2*5)*(2+5)

1125 = 125*(5-2)^(1+1)

1125 = (5^2)*(5*2-1*1)*(5-2+1+1)

1125 = (11+25)*(2^5) - (1+1+25)

1125 = (25-11)*(5*(5-1)^(2*1)) + (5-2+1+1)

1125 = 11*25 + 52*11 + 251 + (5-2)*(5*2-1*1)

1125 = (1+1)^(2*5) + 151 - (52-1-1)

1125^19 + 1125^17 + 1125^15 + 1125^13 + 1125^11 + 1125^9 + 1125^7 + 1125^5 + 1125^3 + 1125 + 1 is prime. (What's the next number with the same property?)

329th Day of the Year.

329 = 293 + (9-3)^2

329 = 239 + (9-3)*(9*2-3)

329 = 392 - (3^2)*(9-2)

329 = 3*29 + 32*9 - 9*(3+2) - (9/3-2)

329 = 9*23 + 92*3 - 239 + (3+2)*(9+2^3)

329 = 2*39 + 93*2 + 3*2*9 + (2+9)*(3-2)

329 = (3*2*9)*((9/3)*2) + (9/3+2)

329 = 23*(3+2+9) + (2^3-(9/3-2))

329 = (32+9)*(9-3+2) + (9-2^3)

329 = (29-3)*((9-3)*2) + (9+2^3)

329 = (3+29) + (32+9) + (9+23) + (92+3) + 3*2*9 + ((3+2)*(9*2-3))

329 + (3+2+9) = 343

329 + (3*2*9) = 383 (both palindromes)

(What's another number with this property?)

329 = 7*47

7+47 = 54 = 3*2*9

(Are there other numbers with this property?)

329 = 7*47

747*329 uses 2,3,4,5,6,7 once (245763). (Do other numbers have a similar property?)

There are 329 primes less than 47^2 (Note: 47 is a prime factor of 329). (Do other numbers satisfy both these properties?)

329 is not divisible by 3,2, or 9. (What's another composite number with this property?)

329^2 + 329 + 1 is prime. (What's the next number with this property?)

There are 329 primes of the form x^16 + 1 less than 10^65.

329 = 107 + 109 + 113 (3 consecutive primes).

###
November 26th (330th Day of the Non-Leap Year)

Nov 26, 11/26.

1126 = 621 + 261 + 216 + (1+1+26)

1126 = 612 + 211 + 162 + 126 + (6-1)*(1+2)

1126 = 611 + 161 + 121 + 116 + 112 + 12-(1+6)

1126 = 112*6 + 11*26 + 1*126 + (6+1)*(2+1)*(6/2-1/1)

1126 = 62*11 + 112*6 - 216 - 1*1*2*6

1126 = 26*11 + 6*211 - 611 + 161 + (1+1+2)*6

1126 = 11^(6/2) - 216 - (1*1-2*6)

1126 = (1*1*2*6)*(62+2^(6-1)) - (12/(1*6))

1126 = (11+26)*(26+(1+1+2)) + 2^(6-1-1)

1126 = (26-11)*(62+11) + 6^2 - (6/2+1+1)

1126 = 11*126 - 26*(1+1+2+6)

1126!!!!! + 1 is prime. (What's the next number with the same property?)

2^1126, 3^1126, 5^1126, 7^1126, 11^1126, and 13^1126 all have even digit sums. (What's the next number with this property?)

1126^4 + 1 is prime (a twin prime with 1126^4 + 3). (What other numbers satisfy both these properties?)

330th Day of the Year.

330 = 303 + 3^3

330 = 33*(3*3+3^0)

330 = (3+3^0)*(3*30) - (3+3)*(3+(3+3^0))

330 = (3*3)*(3+3)^(3-3^0) + (3+3+0)

330 = (3+3+0)*((30-3)*(3-3^0)) + (30/3-(3+3^0))

330 = (30-3)*(3*(30/3-(3+3+0))) + (3+3+0)

330^16 + 1 is prime. (What's another number divisible by 10 with this property?)

330 = 43 + 47 + 53 + 59 + 61 + 67 (six consecutive primes).

There are 330 dimples on a British golf ball.

###
November 27th (331st Day of the Non-Leap Year)

Nov 27, 11/27.

1127 = 711 + 271 + 127 + (1+1)*(2+7)

1127 = 721 + 217 + 17^(1*2) - ((11+2)-(7-2-1-1))^(1+1)

1127 = 712 + 211 + 171 + 11*(7-2-1-1)

1127 = 112*7 + 7^(1*1+2)

1127 = 7*211 - 271 - (11+2)*(7-2) - 1*1*2*7

1127 = 11*27 + 72*11 + (11+27)

1127 = 1*127 + 11*27 + 112*7 - (7+2)^(1+1)

1127 = (1*1*2*7)*((7+2)^(1+1)) - (7+2-1-1)

1127 = (1+1+2+7)*(112-7) - (1*1+27)

1127 = (11+27)*(1+1+27) + (27-1-1)

1127 = (11+2+7)*((1+1)*27) + (27+(11+2+7))

1127 = 7*7*23 (only prime digits used in prime factorization). (What's the next number with this property?)

1127 = 7*7*23

1127*(7+7+23) = 41699 (only has nonprimes when multiplied by its prime factors). (Do other numbers have this property?)

If 1127 = n, 987654321n is prime (Meaning, 9876543211127 is prime). (What's the next number with the same property?)

Only one number in {1,2,3,4,5,6,7,8,9} can go in front of 1127 to make a prime. (Which number is it?)

331st Day of the Year.

331 = 313 + (3/3+1)*(3*3*1)

331 = 133 + (33/(3*1))*((3*3-1)+(13-3))

331 = (3*3*1)*((3+3)^(3-1)) + (3+3+1)

331 = (3+3+1)*(33+13) + (3*3*1)

331 = 33*1 + 3*31 + 1*33 + 13*3 + 133

331 = (3+31)*(3*3+1) - (3*3*1)

331 = (3+31) + (33+1) + (1+33) + (13+3) + 133 + (3+3-1)*((3-1)^(3+1))

331 = (13+3)*(3*(3+3+1)) - (3+3-1)

331 is the 67th prime (67 is prime). 67 is the 19th prime (19 is prime). (What's another prime like this?)

331 is prime and (3+3+1) is prime. (What's another prime with this property?)

10^28 + 331 is the first prime after 10^28.

331^2 - 331 - 1 is prime. (What's the next prime with this property?)

The sum of the first 15 semiprimes is 331.

331 = 59 + 61 + 67 + 71 + 73 (five consecutive primes).

Sound travels just over 331 meters per second in dry air at 0 degrees Celsius.

331, 3331, 33331, 333331, 3333331, 33333331 are all prime.

133, 1333, 13333, 133333, 1333333, 13333333 are all composite.

(Do other primes satisfy this interesting property?)

83^331 - 82^331 is the only known exponent for this number to be a prime (further, this is only a probable prime).

331 is the smallest prime that is Cuban, good, lucky, and happy.

###
November 28th (332nd Day of the Non-Leap Year)

Nov 28, 11/28.

1128 = 811 + 281 + (8-2)^(1+1)

1128 = 821 + 218 + (82+1+1) + (8-2-1*1)

1128 = 812 + (8/2+1+1)^(8/2-1/1) + (2+8)^(1+1)

1128 = 112*8 + 218 + 1*1*2*8 - (8/2-1-1)

1128 = 11*28 + 82*11 - 82*1*1

1128 = 1*128 + 11*28 + 112*8 - 211 + (11-8/2)

1128 = (1+1)^(2+8) + 128 + 8*(2+1*1)

1128 = (1*1*2*8)*(82-11) - (8+2-1-1)

1128 = (1+1+2+8)*(82+11) + 2*(8-1-1)

1128 = (11+28)*(1+1+28) - (8/2+1+1)*(11-8/2)

1128 = (28-11)*(11*(8-2)) + (8-2*1*1)

1128 = 2*2*2*3*47

1128*222347 and 1128*473222 both end in the same three digits (416). (Note both six digit numbers are concatenations of is prime factors in decreasing and increasing order. What's another number, if any, with this property?)

1128 = 2*2*2*3*47 (2 â‰¤ 2 â‰¤ 2 â‰¤ 3 â‰¤ 47). (Are there other numbers with a similar property?)

1+2+3+...+47 = 1128 (Note: 47 is the largest prime factor of 1128).

There are 1128 5-digit numbers N where the reversal of N divides N.

7*(1128^1128) - 1 is prime (1128 is the smallest number to satisfy this property for a 7).

1128 = 1*3*5 + 3*5*7 + 5*7*9 + 7*9*11

There are 5 primes between 112800 and 112899. (Can you name them? All five?)

332nd Day of the Year.

332 = 233 + 33*(2+3/3)

332 = 323 + (3+3+2+(2-3/3))

332 = (3+3+2)*(23+3*3*2) + (3+3-2)

332 = (3*3*2)^((3/3)*2) + (3+3+2)

332 = 33*2 + 3*32 + 23*3 + 2*33 + (33+2)

332 = (3+32) + (33+2) + (2+33) + (23+3) + 233 - 2^(3+2)

332 = 2*2*83

332*(2+2+83) uses only even digits (28884). (Are there others like this?)

The sum of the first 332 primes is prime.

332^32 + 1 is prime. (What's the next number with the same property?)

There are 332 primes less than 10^4 with at least one 6.

2^332, 3^332, and 5^332 all have an even digit sum.

The sum of the proper divisors of 332 is a perfect square (and fourth and eighth power).

332 written in base 5 and 6 differ by 1000 (2312 and 1312 respectively). (Do other numbers have a similar property?)

332 written in bases 19,20,...35 all have at least one letter in them. (What's the next number with this property?)

###
November 29th (333rd Day of the Non-Leap Year)

Nov 29, 11/29.

1129 = 911 + 211 - (11-2*9)

1129 = 912 + 219 - (9/(2+1)-1)

1129 = 921 + 211 - (9/(2+1*1))

1129 = 1*129 + 11*29 + 112*9 - 291 - (1+1+2)*9

1129 = 92*11 + 119 - (9/(2+1)-1)

1129 = 112*9 + (9+2)^(1+1)

1129 = 9*211 - 911 + 129 + (1*1+2+9)

1129 = (1*1*2*9)*(9*(2*9-11)) - (9-2-1-1)

1129 = (1+1+2+9)*(92-(9-2-1-1)) - (9/(2+1)-1)

1129 = (11+29)*(29-1-1) + (9-2)^(1+1)

1129 = 11*129 - 291 + (9^(1/2)-1-1)

1129 = 29*(11+29) - (1+1+29)

1129 = 92*(1+1+2+9) - (92-11) + (2*(9-1-1))

1129 is the smallest prime where the next prime is 22 more than it.

Saying '1129' digit-by-digit ("one one two nine") spells out the number 199, also a prime. (What's the next prime with this property?)

1129 is prime.

1129^4 + 1129^3 + 1129^2 + 1129 + 1 is prime.

(What other primes have this property?)

The reverse of 1129^2 is prime (1464721).

The reverse of 1129^3 is prime (986909341).

(Are there other numbers that have a similar property?)

11^1129 - 10^1129 is prime. (What's the next number with this property?)

1129 written in bases 6 and 8 have the same digits (5121 and 2151, respectively). (Does this happen for other numbers?)

333rd Day of the Year.

333 = 3*(33*3+(3+3+3+3))

333 = (33*3)*(3+3/3) - (33+(33-3))

333 = (3*3*3)*(3+3+3+3) + (3^3)/3

333 = (33+3)*(3+3+3) + (3+3+3)

333 = (3+3+3)*(33+3+3/3)

333 = (33-3)*(33/3) + (3+3-3)

333 = ((3^3)*3)*(3+3/3) + (3+3+3)

There are 333 perfect squares less than 48^3.

333 is a palindrome.

The sum of its proper divisors is also a palindrome (161). (What other palindromes have this property?)

###
November 30th (334th Day of the Non-Leap Year)

Nov 30, 11/30.

1130 = 1310 - 330 + 30*(1+1+3+0)

1130 = 311 + 131 + 301 + 310 + 11*((11-3)-(3-1-1))

1130 = 1031 + 3*(3*11)

1130 = 3011 - 1301 - 311 - 301 + (31+1)

1130 = (11+3)*(3^(3+1*1)) - (3+1*1)

1130 = (11+30)*(3^(1*1*3)) + (11+(13-1-0))

1130 = (1+1+3+0)*(301-(1+1+3+0)*(30/(1+1)))

1130 = (30-11)*((1+1)*30) - (1+1+3+0)*(3-1*1)

1130 = 11^(3+0) - 310 + 101 + (11-3)

1130^32 + 1131^32 is prime. (What's the next number with the same property?)

334th Day of the Year.

334 = 343 + (3-3*4)

334 = 433 - 33*((4-3)*3)

334 = 33*4 + 3*34 + (3+3+4)^(3+3-4)

334 = 4*33 + 43*3 + (33+(43-3))

334 = (3+34) + (33+4) + (4+33) + (43+3) + (3*3+4)^(3+3-4) + ((33-(4-3))/4)

334 = (33+4)*(3*4-3) + ((3+3+4)+(3-3*4))

334 = (3*3*4)*(3*4-3) + (3+3+4)

334 = (33-4)*((33/(4-3/3))) + (3+3*4)

334 = (34-3)*(3+3+4) + 3*(4+3+3/3)

334 = (43+3)*((4*3)-(4+3/3)) + (3*3+4-3/3)

334^4 + 1 is prime. (What's another number with the same property?)

If 334 = n, then the concatenation 2n3n5n7n11n13 is prime (Meaning, 23343334533473341133413 is prime). (What's another number with this property?)

334^5 + 334^3 + 334 + 1 is prime. (What's the next number with this property?)

334 = 2+3*5+7*11+13*17+19

334 written in bases 13 and 14 have the same digits (1C9 and 19C, respectively). (What other numbers have a similar property?)

###
December 1st (335th Day of the Non-Leap Year)

Dec 1st, 12/1.

121 = 11^2 (uses 1,2,1).

121 = 1*21 + 12*1 + 11*(12-(1+2+1))

121 = 12^(1*2*1) - (12+11)

121 = 12*(12-1) - (12-(1+2)+1)

121 = 21*(1*2+2*1+(1*2*1)) - (1+2)! - 1!

121 = (1+2+1)*(12+21) - 11

121 = 3^4 + 3^3 + 3^2 + 3^1 + 1 (Only square number of this form, given 3 is prime).

DigitSum(121^1) = 4^1.

DigitSum(121^2) = 4^2.

121 and 121^2 are palindromes.

121^2 - 121 - 1 is prime. (What's another palindrome with this property? Square?)

121^1 contains 2 different digits.

121^2 contains 3 different digits.

121^3 contains 4 different digits.

(Are there other numbers with this property?)

121 = 37 + 41 + 43 (three consecutive primes).

121 = 5! + 1 (What's another square with this property? Another palindrome?)

121 written in bases 3, 7, and 8 are palindromes (11111, 232, and 171, respectively). (What's another palindrome with this property?)

121 written in bases 5, 9, 11, 15 and 28 are squares (441, 144, 100, 81, and 49, respectively). (What's another square with this property?)

335th Day of the Year.

335 = 353 - (3*5+3)

335 = 533 - 353 + (33-(5-3))*5

335 = 33*5 + 3*35 + (3*3+5-(3+3-5))*5

335 = 53*3 + 5*33 + (3+3+5)

335 = (3+35) + (33+5) + (5+33) + (53+3) + 33*5

335 = (3+35)*(3^(5-3)) - (5+3-3/3)

335 = (3+3+5)*((3+3)*5) + (3/3)*5

335 = (33-5)*(5*3-3) - (3+3-5)

335 = (35-3)*(3*3+5) - (3*3+(3+5))

335 = (3*3*5)*((33-5)/(5-3/3)) + (35-3*5)

335^3 only has odd digits (37595375). (Do other numbers have this property?)

335 = 5*67 (5 < 67 and 5 < 6 < 7). (What other numbers have this property?)

There are 335 primes of the form x^8 + 1 for x less than 10^4.

The sum of the first 19 composite numbers is 335. (Meaning, 335 = 4+6+8+9+10+12+14+15+16+18+20+21+22+24+25+26+27+28+30).

There are more primes in [335,2*335] than in [335^2,336^2]. (What's the next number with this property?)

There are 67 primes below 335 (335 is divisible by 67). (What's another number with this property?)

###
December 2nd (336th Day of the Non-Leap Year)

Dec 2nd, 12/2.

122 = 12^2 - 22

122 = (12-2)^((2-1)*2) + 22

122 = (22/(1+2/2))^2 + (1-2+2)

122 = 12*22 - 212 + (12*(1+2+2)) + (12-2)

122 = (1+2+2)*(12*2) - (1-2)*2

122 = (1*2*2)*(22+2^(1+2)) + 2*(2-1)

122 = (1+22)*(1+2+2) + 12-(1+2+2)

122 = (12+2)*((1+2)^2) - (1*2*2)

122 = (22-1)*((1+2)*2) - (1*2*2)

122^4 uses 1,2,3,4,5,6. (Do other numbers have a similar property?)

122^1 has 2 different digits.

122^2 has 3 different digits.

122^3 has 4 different digits.

(What's the next number with all three properties?)

122*221 is a palindrome (26962) and 2 < 6 < 9 > 6 > 2.

122^2 + 123^2 is prime. (What's another number like this?)

122^4 + 122^3 + 122^2 + 122^1 + 1 is prime.

122^6 + 122^5 + 122^4 + 122^3 + 122^2 + 122^1 + 1 is prime.

(Do other numbers have both these properties?)

There are 122 primes less than 26^2.

The numerator of 1 + 1/2 + 1/3 + ... + 1/122 is prime. (What is the prime number on the numerator?)

The number of primes between [122,2*122] equals the number of primes between [122,122^2]. (What's the next number with the same property?)

678901234567890123456.... (122 digits) is prime.

122 is the largest known number in which there are no twin primes between 122^2 and 123^2.

336th Day of the Year.

336 = 363 - 3^(6-3)

336 = 3*36 + 33*6 + 6*(6-3/3)

336 = 6*33 + 63*3 - 63 + (3+3+6)

336 = (3+3+6)*(33-6+3/3)

336 = (3*3*6)*((6/3)*3) + (3+3+6)

336 = (33+6)*(36/(6-3+3/3)) - (3*(6-3/3))

336 = (33-6)*(3+3+6) + (36/3)

336 = (36-3)*(6+3+3/3) + (3/3)*6

336 = (3+36) + (33+6) + (6+33) + (63+3) + 3*3*6 + (33/(6-3))*(3*3)

678901234567890123456.... (336 digits) is prime. (Largest number known with this property. Second largest is with 122 digits).

There are 336 primes of the form x^8 + 1 less than 10^32.

336^2 + 337^2 + 338^2 is prime. (What's the next number with this property?)

336^32 + 337^32 is prime. (What's another number with this property?)

There are 336 dimples on an American golf ball.

###
December 3rd (337th Day of the Non-Leap Year)

Dec 3rd, 12/3.

123 = 13^2 - (3-2+1)*23

123 = 12^(3-2+1) - (2^3-1)*21

123 = 12*3 + 1*23 + 3*21 + 32*1 - 31

123 = (12+3) + (1+23) + (3+21) + (32+1) + (1+2)^3

123 = (13-2)^(1-2+3) + (3-2+1)

123 = (1+2+3)*((3+2-1)*(3*2-1)) + (3*(2-1))

123 = (1+23)*(3*2-1) - (1-2)*3

123 = (12+3)*(1*2^3) + (12/3-(3-2)/1)

123 = (12-3)*((2^3-1)+(1*2*3)) + (1+2+3)

1*2*3 = 1+2+3

123^4 only has odd digits (disregarding the ones digit). (Are there other numbers with the same property?)

3^123 + 2 is prime and 123^3 +2 is prime (Note: 3 and 2 used here are also in the string '123'). (Are there other numbers that satisfy both these properties?)

10^123 + 3 is prime (Meaning, 1000...0003 where there are 122 zeroes is prime). (What's the next number with the same property?)

10^25 - 123 is the largest prime less than 10^25.

10^63 - 123 is the largest prime less than 10^63.

123^4 + 124^4 is prime.

(Also, 121^4 + 122^4 and 122^4 + 123^4 are prime).

(What's another run of 3 consecutive numbers satisfying this property?)

337th Day of the Year.

337 = 373 + (3-7)*(3^(3*3-7))

337 = 3*37 + 33*7 - (7-3+3/3)

337 = 7*33 + 73*3 + 3*3*7 + (33+7) + (3+7)

337 = (3+37) + (33+7) + (73+3) + (7+33) - (3*3*7)*(3*3-7) - 3*(7-3+3/3)

337 = (3+3+7)*(33-7) - (7-3-3)

337 = (3*3*7)*(7-3+3/3) + (3*7+3/3)

337 = (33+7)*(7+3/3) + (7*3-(7-3))

337 = (37-3)*(7+3) - (7-3-3/3)

337 = 7^3 - (7-3/3)

337^2 = 113569 (1 â‰¤ 1 â‰¤ 3 â‰¤ 5 â‰¤ 6 â‰¤ 9). (Does this happen for other numbers?)

337 + 3*3*7 = 400 = 20^2.

337 and 733 are prime.

3,7,33,37,337,373,and 733 are the permutations of 337 and all are prime.

2*3*5*7*11*13*17*...*337 - 1 is prime. (When will this happen next? If at all?)

337^19 + 337^17 + 337^15 + 337^13 + 337^11 + 337^9 + 337^7 + 337^5 + 337^3 + 337^1 + 1 is prime. (What's the next prime with this property?)

338^11 - 337^11 is prime. (What's the next number with this property?)

(3!)^(3!) + 7 and (3!)^(3!) - 7 are consecutive primes.

###
December 4th (338th Day of the Non-Leap Year)

Dec 4th, 12/4.

124 = 14^2 - 24*(4-2+1)

124 = 12^(4-2*1) - (12+1*2*4)

124 = (14-(4-2+1))^(4-2*1) + (4-2+1)

124 = 12*4 + 1*24 + (42+(4+1)*2)

124 = 4*21 + 42*1 - (4-2*1)

124 = (12+4) + (1+24) + (4+21) + (42+1) + (4-1)*(4+1)

124 = 214 - 2*(42+(4-2+1))

124 = 142 - (4-2)*(4*2+1)

124 = (1+2+4)*(1+2^4) + (4+2-1)

124 = (1*2*4)*(24-(1+2*4)) + (2-1)*4

124 = (12+4)*(2^(4-1)) - 4*(2-1)

124 = (1+24)*(4+2-1) - (4-2-1)

124^2 + 1 is prime. (What's the next number with the same property?)

The sum of the first 124 primes is prime.

3^124 + 28 and 3^124 - 28 are prime. (What's the next number with this property?)

124 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 (8 consecutive primes).

+- 1 +- 2 +- 3 +- 4 +- 5 +- 6 +- 7 +- 8 +- 9 +- 10 +- 11 +- 12 = 0 has 124 solutions.

124 written in bases 8 and 9 use the same digits (174 and 147, respectively). (What other numbers share a similar property?)

338th Day of the Year.

338 = 383 - (8+3/3)*(8-3)

338 = 3^(8-3) + (83+(8+3+3/3))

338 = (8-3)^3 + (3+3)^(8-3-3+3/3) - (8-3+3/3-3)

338 = 3*38 + 33*8 - 8*(8-3)

338 = 8*33 + 83*3 - 3*3*8 - (83+(8*3-3)-(3*3-8))

338 = (3+3+8)*(33-8-3/3) + (8-3-3)

338 = (3*3*8)*(8-3) - (33-(3+8))

338 = (33+8)*(3-3+8) + (8-3-3)*(8-3)

338 = (33-8)*((38+3/3)/3) + (3+3+8-3/3)

338 = (38-3)*(8+3/3) + (8*3-3/3)

The digits of 338 do not appear in 338^2. (What other numbers have this property?)

There are 338 primes between 35^3 and 36^3.

###
December 5th (339th Day of the Non-Leap Year)

Dec 5th, 12/5.

125 = 5^(2+1)

125 = 152 - (5-2)^(2+1)

125 = 215 - 21*5 + 15

125 = 1*25 + 12*5 + (1+2+5)*(5*(2-1))

125 = 5*21 + 52*1 - 1*2^5

125 = (1+2+5)*((1+2)*5) - (1-2)*5

125 = (1*2*5)*(2*(1+5)) + 5*(2-1)

125 = (1+25)*(5*(2-1)) - ((5+1)/2 + (5-2-1))

125 = (12+5)*(2+5) + (5+2-1)

125 = (52+1)*(5-2-1) + (21-(5-2-1))

125 = 2^(5+1) + (52+(5*2-1))

125^n ends in "25" for all positive integers n. (This works for any number ending in 25).

125*521 contains 125. (What's the next number with the same property?)

125 = 5^3.

1+2+5 = 2^3.

(Are there other numbers with both these properties?)

There are 125 primes less than 700.

125^2 + 4 is prime (Note: 125 is of the form p^2 + 4 for prime p as well). (What's another number with both these properties?)

125 equals the third prime to the third power.

There are 125 primes between 19^3 and 20^3.

There are 39 primes between 125^2 and 126^2 and 39 primes between 125 and 2*125.

339th Day of the Year.

339 = 393 - (3+3)*9

339 = (9-3)^3 + (9/3)^3 + (93+3)

339 = 3*3*9 + (3+3)^(9/3) + (33+9)

339 = 3*39 + 33*9 - (33-(9-3/3))*(9/3)

339 = 9*33 + 93*3 - 393 + (9-3-3-3/3)^(9-3/3-3/3) + (9-3-3/3-3/3)*(9-9/3+3/3)

339 = (3*3*9)*(9/3+3/3) + (3+3+9)

339 = (3+3+9)*(33-(9+3-3/3)) + 9*(3/3)

339 = (3+39)*(9-3/3) + (9-3-3)

339 = (33-9)*(9+3+3-3/3) + (9/3)

339 = (39-3)*((3/3)*9) + (3+3+9)

339^10 + 339^9 + 339^8 + 339^7 + 339^6 + 339^5 + 339^4 + 339^3 + 339^2 + 339^1 + 1 is prime. (What's the next number with this property?)

If 339 = n, 2n3n5n7n11n13 is prime (Meaning 23393339533973391133913 is prime).

###
December 6th (340th Day of the Non-Leap Year)

Dec 6th, 12/6.

126 = 2^(1+6) - (6/2-1)

126 = 6^(2+1) - (62+21) - (6+2-1)

126 = 162 - 6^(2*1)

126 = 261 - ((1+2)^(6/2))*(6-(2-1))

126 = (1*26) + (12*6) + (6+2-1)*(6/2+1)

126 = (6*21) + (62*1) - 2*(26+(6-(2-1)))

126 = 612 - 261 - (12+(6-2-1))^(6/2-1)

126 = (1+26) + (12+6) + (6+21) + (62+1) - (1+2+6)

126 = (1+2+6)*(16-2)

126 = (1*2*6)*((6-1)*2) + (6*(2-1))

126 = (12+6)*(6+2-1)

126 = (1+26)*(6/2+1) + (1+2)*6

126 = (62+1)*(6/2-1)

126^3 contains 3 zeroes yet 126 has no zeroes. (126 is the smallest number with this property.)

126^2 + 1 is prime.

Nuclei with 126 protons or neutrons are the largest nuclei that are more stable against nuclear decay.

126^6 + 126^5 + 126^4 + 126^3 + 126^2 + 126^1 + 1 is prime. (What's the next number with this property?)

There are 126 perfect squares less than 25^3 (incl 25^3).

340th Day of the Year.

340 = 304 + (40-3-(4-0-3))

340 = 4^3 + 3^4 + (40/((4+3)+(4-3)))*(34+(30/((4+3)-(4-3))))

340 = 403 - 3^4 + 3*(3+0+4-(4-3))

340 = 430 - 3*(40-(4+3+3))

340 = (3+4+0)^3 - 3*(4-3-0)

340 = 3*40 + 34*(3+0+4) - 3*((4+3)-(4-3))

340 = 4*30 + 3*40 + (3+4+((34-(4-3))/((4+3)+4*(4-3))))^((4-3)-(3-4))

340 = (34+0)*(3+4+3*(4-3))

340 = (3+40)*((4+3)+(4-3)) - (4*(4-3))

340 = (3+4+0)*(4*(3*4)) + (4*(4-3))

340^2 + 1 is prime.

340! + 1 is prime.

340^4 + 1 is prime.

(What's another number with two or all three of these properties?)

There are 340 ways to place two nonattacking queens on a 6 by 6 board.

There are 340 ways to place 5 nonattacking knights on a 4 by 4 board.

340 = 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 (8 cons primes).

340 = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 (10 cons primes).

###
December 7th (341st Day of the Non-Leap Year)

Dec 7th, 12/7.

127 = 172 - (1*7+2)*(1*7-2)

127 = 217 - (7+2)*(1+2+7)

127 = (1*27) + (12*7) + (12+(7-2-1))

127 = (7*21) + (72*1) - (7-2-1)*(17+(7-2+1))

127 = 2^(7*1) - (7+2+1) - (1+2)*(7-2-1-((7-2)/(12-7)))

127 = (12+7) + (1+27) + (7+21) + (72+1) - (1+2)*7

127 = (12+7)*(7*(2-1)) - (7-(2-1))

127 = (12-7)*((7-2)^(2-1)) + (1+2+7)/(7-2*1)

127 = (1+27)*(7/2+1) + (12/(7*2-2*(2-1)))

127 = (72+1)*((7-2-1)/(2*1)) - (17+2)

127 = (27-1)*(7-2*1) - (7-2-1-(2-1))

127 = (1+2+7)*(7*2-1) - 12 - (7+2*1)

127 = (1*2*7)*(1*2+7) + (17-2^(7-2-1))

127^4 + 127^3 + 127^2 + 127^1 + 1 is prime. (What's another number like this?)

127^16 + 128^16 is prime. (What's the next number satisfying the same property?)

127 is a Mersenne prime (prime of the form 2^p - 1 for a prime p).

Further, 2^127 - 1 is a prime itself. It is the largest prime ever discovered by hand calculations.

127 = 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0.

There are 127 prime days in a leap year.

341st Day of the Year.

341 = 314 + 3^(4-1)

341 = 413 - (3^(4-3+1))*(4^(3-1)-(3+4+1))

341 = 3*41 + 34*1 + 14*3 + 1*43 + 3*(34-1)

341 = (34+1) + (3+41) + (1+43) + (14+3) + 14^(3-1) + (4+1)

341 = (34+1)*((4+1)*(4-3+1)) - 3*(4-1)

341 = (34-1)*((3*4)-(4-3+1)) + (3*4-1)

341 = (3+41)*(31/4)

341 = (1+43)*(3+4+1) - (3*4-1)

341 = (14+3)*(3*4*1+(3+4+1)) + ((4-1)/3)

341 = (3+4+1)*(43-1) + (3+4+1-(4-1))

341 = (3*4*1)*(31-4) + (14+3)

341 = 4^4 + 4^3 + 4^2 + 4^1 + 4^0.

341 = 37 + 41 + 43 + 47 + 53 + 59 + 61 (7 cons primes).

341^6 + 341^5 + 341^4 + 341^3 + 341^2 + 341^1 + 1 is prime. (What's the next number with the same property?)

There are 341 ways to place 11 nonattacking queens on an 11 by 11 board.

341 written in bases 2, 4, 8, 17, and 30 are palindromes (101010101, 11111, 525, 131, and BB, respectively). (What other nonpalindromic numbers are nontrivial palindromes in 5 bases?)

###
December 8th (342nd Day of the Non-Leap Year)

Dec 8th, 12/8.

128 = 2^(8-1)

128 = (1+2+8)^(12-(1*2+8)) + (8+1-2)

128 = 182 - (8+1)*(8-2)

128 = 218 - (82+8*(2-1))

128 = (8-2)^(2+1) - (81+(8-2+1))

128 = 1*28 + 12*8 + 8/(2*1)

128 = 8*21 - 82*1 * (8-1)*(8-2)

128 = (1+28) + (12+8) + (8+21) + (82+1) - (1+2)*(1+2+8)

128 = (1+2+8)*((8/2-1)*(8/2)) - (8/(2*1))

128 = (1+28)*(8/2*1) + (8-2)*(2/1)

128 = (12+8)*(8-2/1) + 8*(2-1)

128 = (1*2*8)*(8/2*1)

128 = 12^((18/2)-(8-2+1)) - 1*2*8

128 = 2^7. 1*2*8 = 2^4.

The sum of divisors of 128 is also a power of 2 (64 = 2^6).

(Do other numbers have all these properties?)

128^6 + 128^5 + 128^4 + 128^3 + 128^2 + 128^1 + 1 is prime.

128 written in binary (base 2), quaternary (base 4), and hexadecimal (base 16) begins with 1, 2, and 8 (Note the bases; 2 is two times 1, 4 is two times 2, 16 is two times 8).

128 is the largest number that cannot be expressed as the sum of any number of distinct squares.

The Commodore 128 home/personal computer which had 128 KB of memory was released in 1985.

342nd Day of the Year.

342 = 243 + 3*(42-(3+4+2))

342 = 234 + (3*4)*(3+4+2)

342 = 324 + 3*(4+2)

342 = ((3+4+2)*2)^(2*(4-3)) + (34+2)/(2*(4-3))

342 = 423 - 3^(4*(3-2))

342 = 432 - 2*(42+3)

342 = (3+4+2)*((42-3)-(3-(4-2))

342 = 3*42 + 34*2 + 2*43 + 24*3 - (3*4-2)

342 = (34+2) + (3+42) + (2+43) + (24+3) + (3*4+2)^(2*(4-3)) - (3*4/2+(2-(4-3)))

342 = (34+2)*((3+4^2)/(2*(4-3)))

342 = (3+42)*((34+4*(3-2))/(3+4-2))

342 = (24+3)*(3^2+4) - (3+4+2)

342 = (3*4*2)*(3*4+2) + (3*4)/2

342 = ((3*4/2)+2-(4-3))^(2+(4-3)) - (3-(4-2))

342^2 is the concatenation of four squares (1//16//9//64). (Note: 4 is 2-squared. Are there others with this property?)

342^3 has 3 zeroes (yet no zeroes are in 342).

There are 342 primes less than 2300.

There are 342 primes less than 48^2.

The sum of the first 342 primes is prime. (What's the next number with this property?)

342^32 + 1 is prime. (What other numbers have this property?)

The concatenation of the first 342 primes is prime. (What other numbers satisfy this property?)

There are 342 primes between 49 and 49^2.

342^2 + 343^2 + 344^2 is prime. (What's another number with the same property?)

342^343 + 343^342 is prime. (What's another number with this property?)

###
December 9th (343rd Day of the Non-Leap Year)

Dec 9th, 12/9.

129 = 2^(9-(2*1)) + (12-9) - (12/(9-2-1))

129 = 219 - 9*(9+2-1)

129 = (1*2+9)^(29-(12-9)^(9/(2+1))) + (9+1-2)

129 = 192 - (9*(2-1))*(9-2*1)

129 = 1*29 + 12*9 - (9-(2-1))

129 = 9*21 - 92*1+ 2^((1+9)/2)

129 = (1+29) + (12+9) + (92+1) + (9+21) - 9*((12-9)+(12/(9-2-1)))

129 = (1+2+9)*(9+2*1) - (9/(2+1))

129 = (1*2*9)*(9-2*1) + (12-9)

129 = (1+29)*(12/(12-9)) + 9*(2-1)

129 = (12+9)*(9-2-1) + (9/(2+1))

129 = 3*43 (Today is the 343rd Day of the Year).

129^2 is the concatenation of 3 squares (16//64//1).

129^4 + 129^3 + 129^2 + 129 + 1 is prime. (What's the next number like this?)

2*129 + 19 and 2*129 - 19 are prime (Note the use of 2 and 19, digits in 1,2,9). (Are there other numbers that share a similar property?)

There are 129 primes less than 27^2 (or 3^6, a stronger fact).

129^8 + 130^8 is prime.

129^4 + 130^4 is prime.

(Do other numbers have both properties?)

The number of primes between 129 and 129^2 equals the number of primes between 129 and 2*129. (What's another number with the same property?)

The sum of the first 10^1 primes is 129.

129 is the smallest number that can be expressed as the sum of three squares in four different ways.

343rd Day of the Year.

343 = (3+4)^3

343 = 433 - 3*((3+4*3)*(3+3-4))

343 = 334 + (3*4-3)

343 = 3*43 + 34*3 + 4*(33-4-(4-3))

343 = (3+43) + (34+3) + (33+4) + (3*4+3)^(3+3-4) + 3-(4-3)

343 = (3+4+3)*34 + 3*(4-3)

343 = (34+3)*(3*4-3) + (3+4+3)

343 = (34-3)*(3*4-(3/3)) + (3-4+3)

343 = (3+43)*(3+4) + 3*(4+3)

343 = (3*4*3)*(3+4+3) - 3*(4+3) + (4*(3/3))

There are 343 primes less than the product of the first 5 primes (i.e. 343 primes less than 2*3*5*7*11 = 2310).

If 343 = n, n0123456789 is prime (Meaning, 3430123456789 is prime). (What's another number with this property?)

343 is the first nontrivial cubic palindrome. (What's the next one? Is there one?)

The speed of sound in dry air at 20 deg C is 343 m/s.

343 NYC Fire Department members died due to the events caused on September 11, 2001. May they rest in peace.

343 written in bases 6, 11, and 18 are also palindromes (1331, 292, and 111, respectively).

343 written in bases 6 and 7 are also perfect cubes (1331 and 1000, respectively).

###
December 10th (344th Day of the Non-Leap Year)

Dec 10th, 12/10.

1210 = 11^(2+2^0) - (12-1)^(1*2*1)

1210 = 1120 + (1*2*1)*(12-(1+2))*(10/(1*2))

1210 = 1102 + 10^(1*2) + (10-1*2)

1210 = 1201 + (1+2)^(1*2)

1210 = (1+2+1+0)*(201+101+(1/2))

1210 = 12*210 - 1201 - (121-12)

1210 = 121*10

1210 = (12+10)*(11*(10/(1*2)))

1210 = (12+1)*(10^2 - (10-(1+2))) + ((1*2)/(1+0))

1210^2 + 1 and 1210^2 + 3 are twin primes. (What's another number with the same property?)

1210 has one of every digit in some base.

1210 base 5 uses all possible digits (14320).

344th Day of the Year.

344 = 434 - (3*(4/4))*(34-4)

344 = 443 - 3*(44-(4+4+3))

344 = (3+4)^(3*(4/4)) + (4-3)

344 = (4+4)^3 - (3*4)^(3-4/4) - 3*(4+4)

344 = (3*44) + (34*4) + (4*(3+4*4))

344 = (4*43) + (44*3) + (4*(3+4+4-4/4))

344 = (3+44) + (34+4) + (4+43) + (44+3) + (3*4+4/4)^(3-4/4) - (3+4/4)

344 = (3+44)*(3*4-4) - (3-4/4)^(4+(4-3))

344 = (34+4)*(3^(3-4/4)) + (3-4/4)

344 = (3+4+4)*(34-3*(4/4)) + (3*(4/4))

344 = (3*4*4)(4*4-(3^(3-4/4))) + (3-4/4)^3

There are 344 ways to place 4 nonattacking queens on a 4 by 8 board.

There are more primes between 344 and 2*344 than 344^2 and 345^2. (What's the next number with the same property?)

345^11 - 344^11 is prime. (What's another number with this property?)

If 344 = n, n0123456789 is prime (Hence, 3440123456789 is prime). (What's the next number with this property?)

The most common time to wake up during the night is 3:44 AM.

###
December 11th (345th Day of the Non-Leap Year)

Dec 11th, 12/11.

1211 = 11^(2+1) - (1+2+1+1)!

1211 = 12*211 - 1121 - 2*(12-(1+1))^(1*2*1*1)

1211 = 12^(1+1+1) - (2^(1+1+1))^(12/(1+2+1)) - (1+2+1+1)

1211 = 112*21 - 1121 - (21-(1*1))

1211 = (12+11)*((12+1*1)*(1+2+1*1)) + (1+2)*(1+2+1+1)

1211 = (1+2+1+1)*(2*(11^2)) + (12-11)

1211 = (1+211) + (12+11) + (121+1) + (11+21) + (112+1) + 1*211 + 12*11 + 121*1 + 12^(1+1) + (111-2*(1+2+1+1))

1211 = (1+211)*(2*(1+1+1)) - 12*(1+2+1+1) - (12-11)

1211 = (121+1)*(12-(1+1)) - (1+1+1)^2

1211*1121 = 1357531 (ascending palindromic number with only odd digits). (Is there a number with the property that produces 2468642?)

1211^0 + 1212^1 + 1213^2 + 1214^3 + 1215^4 + 1216^5 + 1217^6 + 1218^7 + 1219^8 + 1220^9 + 1221^10 + 1222^11 + 1223^12 is prime. (What's the next number with this property?)

1 is described by the number 11.

11 is described by the number 21.

21 is described by the number 1211.

345th Day of the Year.

345 = 354 - (3*4-5) - (3-(5-4))

345 = 435 - (4+5)*(5+4+(4-3))

345 = (3*45) + (34*5) + (3+5)*(5*(4-3))

345 = (5*43) + (54*3) + (5-4-3)^(5*(4-3))

345 = (3+45) + (34+5) + (5+43) + (54+3) + (54-3)*(3*(5-4))

345 = (3+4+5)*(34-5) + 3*(4-5)

345 = (3*4*5)*(5+4-3) - (3*4+3*(5-4))

345 = (34+5)*(3^(3-(5-4))) - (5+4-3)

345 = (3+45)*(3*4-5) + 3^(3-(5-4))

345 = (54-3)*(3*4-5) - (3+4+5)

345 = (54+3)*(5+4-3) - 3*(4-5)

345 = 3*5*23 (only uses digits 1,2,3,4,5)

345*(3523) = 1215435 (still only uses 1,2,3,4,5).

345 written in bases 8,17 and 11,12 use the same digits in a different order.

###
December 12th (346th Day of the Non-Leap Year)

Dec 12th, 12/12.

1212 = 1221 - ((2+1)^(2*1))

1212 = 1122 + (1+2+1+2)*(12+1+2)

1212 = 12^(1+2) - (2*2*(1+1))^(1+2) - (1*2*1*2)

1212 = (12-1)^(2+1) - (121-2)

1212 = 12*212 - (21-(12-1*2))^(1+2) - (12/12)

1212 = 21*121 - 1221 - (121-(12-1+2))

1212 = 1*212 + 12*21 + 121*2 + 21*21 + (1+2+2)*(12-1+2)

1212 = (1+2+1+2)*(212-(12-1*2))

1212 = (12+12)*(12*(1+2) + 12+(1+2)) + (12/(1-2))

1212 = (1+212)*(1+2+1+2) - (12/(1*2))*(12-2+1)

1212 = (121+2)*(12-1*2) - (21-12)*2

1212^2 only contains nonprime numbers.

1212^9 + 1212^7 + 1212^5 + 1212^3 + 1212^1 + 1 is prime. (What's the next number with the same property?)

346th Day of the Year.

346 = 364 - (3*4+6)

346 = 436 - 3*(6*(6-(4-3)))

346 = 643 - 364 + (64+3)

346 = (3+46) + (34+6) + (6+43) + (64+3) + ((6-4)*(4-3))*(3*4*6) - (6-3)*(4-3)

346 = (3*46) + (34*6) + (34-6)/(6+4-3)

346 = (6*43) + (64*3) - (3*4*6) - (3*4/6)^(6+3-4)

346 = (3+4+6)*(3^((6-4)+(4-3))) - (6-(4-3))

346 = (34+6)*(3^(6-4)) - (3*6-4)

346 = (3+46)*(6+4-3) + ((6-4)+(4-3))

346 = (3*4*6)*((3+4+6)-(6*4/3)) - (3+4+6) - (3-(6-4))

346 = (64+3)*((6+4)/(6/3)) + ((34-(4-3))/(3*(3-(6-4))))

346^2 does not include a 3,4, or 6 (excl ones digit--note 6^n for any positive integer n will end in a 6).

346 has the property that 3 < 4 < 6.

346*643 = 222478 (22 < 24 < 78).

(What's another number with both these properties?)

3+4+6 = 13.

346 = 2*173 and 2+1+7+3 = 13.

3^346 - 346^3 is prime. (What's the next number with this property?)

###
December 13th (347th Day of the Non-Leap Year)

Dec 13th, 12/13.

1213 = 1321 - 12*(3^2)

1213 = 1312 - 3*(31+2*1)

1213 = 1231 - 2*(3^(1+1))

1213 = 1132 + (3^(1+1+2))

1213 = 12*213 - (12-1)^3 - (13-2+1)

1213 = 31*121 - 2311 - 231 + (1+3)/(2-1)

1213 = 1*213 + 12*13 + 121*3 + 312*1 + 13^(1*2)

1213 = (1+213) + (12+13) + (121+3) + (3+121) + (31+21) + (312+1) + 31*21 - (2*(1+1+3))*(31-2*1)

1213 = (1+213)*(12/(3-1)) - (12+(1-3))*(1+2+1+3) + (12-13)

1213 = (12+13)*((1+2+1+3)^((3-1)*(2-1))) - 3*(1+2+1)

1213 = (121+3)*(2^3+1+1) - (1+2)^(1*3)

1213 = (1+2+1+3)*(213-(31+12-(1+2))) + (3+1)/(2-1)

1213^3 doesn't contain 1,2,3 (excl first digit--the 1 must be there).

1213 is prime.

Saying 1213 digit-by-digit ("one two one three") spells out 23, also a prime.

1213^19 + 1213^17 + 1213^15 + 1213^13 + 1213^11 + 1213^9 + 1213^7 + 1213^5 + 1213^3 + 1213^1 + 1 is prime. (What's another prime with the same property?)

1213 is prime.

"One thousand two hundred thirteen" uses 29 letters (29 is prime). (What other numbers have this property?)

1213 is the smallest of 8 consecutive emirps (primes which, when written backwards still yield a prime).

347th Day of the Year.

347 = 7^3 + 4

347 = 374 - (7-4)^3

347 = 437 - 3*((7-4)*(7+3))

347 = 3*47 + 34*7 - ((7*4)/(3+4+7))^(3*4-7)

347 = 7*43 + 74*3 - (3*4*7) - (73+(3*4+7))

347 = (3+47) + (34+7) + (7+43) + (74+3) + (3*4*7) + (3*4-7)*((7+4-3) + (7-4)/3)

347 = (3+4+7)*(7*4-3) - (3*4)/(7-3)

347 = (3*4*7)*(7-3) + (37-4)/(7-4)

347 = (34+7)*(7+4-3) + (3*4+7)

347 = 7*(3+47) - ((7-3)-(4-3))

347 = (47-3)*(7+4-3) - (3*4-7)

347 = (74-3)*(3*4-7) - (7+4-3)

347 has the property 3 < 4 < 7.

347^2 = 120409 (1 < 2 < 4 < 9, exclude zeroes).

347 is prime.

347 - 3*4*7 = 263 is prime.

347 + 3*4*7 = 431 is prime.

347^2 + 4 is prime. (What's the next prime with the same property?)

(347,349) is the last twin prime pair of the year.

The number 111...111 (347 ones) has only two factors. (What factors are they?)

If 347 = p, 987654321p is prime (Meaning 987654321347 is prime). (What's the next prime with this property?)

347 is prime.

347 has 7 syllables (7 is also prime).

Inserting the same digit between the digits of 347 produces just one prime. (Which number must be inserted (i.e. 3_4_7) in order to produce a prime?)

Strobogrammatic numbers do not contain 3,4, or 7. (Strobogrammatic numbers are numbers that, when flipped, still produce numbers).

347 is prime if you add 2 to any of its digits (547, 367, and 349 are all prime). (Are there other numbers with the same property?)

There are exactly 347 even digits before the 347th odd digit of pi.

There were 347 minesweepers that helped the D-Day invasion in World War II.

###
December 14th (348th Day of the Non-Leap Year)

Dec 14th, 12/14.

1214 = 1412 - 214 + (1*2)^(1*4)

1214 = 1241 - (4-1)^(2+1)

1214 = 1421 - 211 + (4*1)/(2-1)

1214 = 1142 + (1+2)*(24*1*1)

1214 = 1124 + (12+(4-1))*(1*2+1*4)

1214 = 1*214 + 12*14 + 121*4 + 421 - 12*(2+4) - ((4-1)/(2+1))

1214 = 41*21 + 412*1 - (41+14+(1+2+1))

1214 = (1+214) + (12+14) + (121+4) + (41+21) + (412+1) + 421 - 12*(1*4)

1214 = (1+2+1+4)*(141+2) + 2*(21+14)

1214 = (12+14)*(41+(1*2+1*4)) - (1*2*1*4)

1214 = (1+214)*(2+4) - 4*(2^4+(4-1)*(2-1))

1214 = (121+4)*(1+1+2*4) - (2+4)^(1+1)

348th Day of the Year.

348 = 384 - (8*3/4)^(8/4)

348 = 438 - 3*((4*8+(4-3))-3)

348 = 843 - 483 - (3*(8-4))

348 = 3*48 + 34*8 - 4*(8*3-(8+3-4))

348 = 8*43 + 84*3 - 384 + 3*4*8 + 8*(3+8/4)

348 = (3+48) + (34+8) + (8+43) + (84+3) + 3*4*8 + 3*(8-4+3)

348 = (3+4+8)*(34-(8+3)) + (48/((3*4-8)^(8/4)))

348 = (3*4*8)*(4*(8-4-3)) - (48-(4+8))

348 = (3+48)*(8+3-4) - 3^(8/4)

348 = (34+8)*(8*(4-3)) + (3*(8-4))

348 = (84-3)*(8-4) + (3*4*8)/(3+(8-4-3))

348 = (84+3)*(3*4-8)

348 + 3*4*8 = 444 (palindrome).

348 - 3*4*8 = 252 (palindrome).

348 + (3+4+8) = 363 (palindrome).

348 - (3+4+8) = 333 (palindrome).

(What's another number satisfying all these properties?)

348 (3 < 4 < 8). 348 = 29*3*2*2.

348*29322 = 10204056 (1 < 2 < 4 < 5 < 6).

(Are there other numbers with the same property?)

348^16 + 1 is prime.

348^8 + 349^8 is prime.

(Do other numbers have both these properties?)

348 written in bases 15 and 17 have the same digits (183 and 138, respectively).

###
December 15th (349th Day of the Non-Leap Year)

Dec 15th, 12/15.

1215 = 1125 + 5*(12+1+5)

1215 = 1251 - (5+1)^(2*1)

1215 = 1512 - 251 - (51-5*1)

1215 = 1152 + (1+2+1+5)*(12-1*5)

1215 = 1*215 + 12*15 + 121*5 + 5*(12+21+1*2*1*5)

1215 = 51*21 + 512*1 - 511 + 12^(5-1-2*1) - (5-1-2-1)

1215 = (1+215) + (12+15) + (121+5) + (5+121) + (51+21) + (512+1) + 5*((1+2)^(2+1))

1215 = (12+15)*(51-(5+1))

1215 = (1+215)*(5*1+(2-1)) - (1+2)^(5-1)

1215 = (121+5)*(1*2*1*5) - 5*((5-2)^(1+2))

1215 = (1+2+1+5)*(215-((1*2)^(5-1))*5)

1215 = (1*2*1*5)*((12-1)^(5-2-1)) + (5*(2-1))

If 1215 = n, 7nn7 is prime (Meaning 7121512157 is prime). (What's another number with this property?)

1215 written in bases 6 and 7 use the same digits (5343 and 3354, respectively).

349th Day of the Year.

349 = 394 - (39+(9-3))

349 = 439 - 9*(9+4-3)

349 = 3*49 + 34*9 - 3*4*9 + (4*(4-9/3))

349 = 9*43 + 94*3 - 394 + (49+(9-4)^(9-4-3))

349 = (3+49) + (34+9) + (9+43) + (94+3) + 3*4*9 - (3*4-9)

349 = (3+4+9)*((9-4-3)*(43-(9-4-3)^(9-4)) - (3*4-9)

349 = (3*4*9)*(3*4-9) + (3*4-(3+4))^(9-4-3)

349 = (3+49)*(4+9/3) - (9-4)*(9/3)

349 = (34+9)*(9-(4-3)) + (3*4-(3+4))

349 = (94-3)*(((9+3)/4)+(4-3)) - (4+9+(9-4-3))

349 = (34-9)*((9+3)+(9-4-3)) - (4-9/3)

349 is prime.

349 + 3*4*9 is prime.

349 - 3*4*9 is prime.

(What's the next prime with this property?)

349^6 + 349^5 + 349^4 + 349^3 + 349^2 + 349^1 + 1 is prime.

United States President Barack Obama had 349 electoral votes in the final election projection of 2008.

###
December 16th (350th Day of the Non-Leap Year)

Dec 16th, 12/16.

1216 = 1261 - ((1+2+6)*(6-1))

1216 = 1162 + (61-(6+1))

1216 = 1126 + 6*(16-(2-1))

1216 = 1612 - 261 - 5*((1+2)^(6/2))

1216 = 1*216 + 12*16 + 121*6 + (61+21)

1216 = 61*21 + 612*1 - 621 - (6+2)*(6+2-1*1)

1216 = (1+216) + (12+16) + (121+6) + (6+121) + (61+21) + (612+1) + (16+1*6)

1216 = (1+2+1+6)*((12-1)^(6/2-1)) + ((6*1)*(2-1))

1216 = (1*2*1*6)*(112-(12-1)) + (6*1)-(2*1)

1216 = (12+16)*(61-6*(1+2)) + (1*2*1*6)

1216 = (1+216)*(6*(2-1)) - (61+(6-1)^(2*1))

1216 = (121+6)*(1+2+1+6) - (61-(6+1))

1216^23 + 1216^21 + 1216^19 + 1216^17 + 1216^15 + 1216^13 + 1216^11 + 1216^9 + 1216^7 + 1216^5 + 1216^3 + 1216^1 + 1 is prime. (What's the next number with the same property?)

There are 1216 composites less than 38^2 (and incl 38^2).

There are 1216 cousin primes (primes differing by 4) less than 10^5.

1 + the product of the first 1216 composite numbers is prime (1216 is the largest known number to have this property).

350th Day of the Year.

350 = 305 + 5*(3^(5-3))

350 = 3*50 + 5*30 + 5*(5+3+(5-3))

350 = (3+50) + (35+0) + (5+3) + (0+53) + 3*50 + (15 + (5+3^0)^(5-3-0))

350 = (50-3)*(5+3-5^0) + 3*((5+3)-3^0)

350 = (50+3)*(3*(5-3-0)) + (5-3)^(5*3^0)

350 = 35*(3+5-(3-5))

350 = 2*5*5*7

2+5+5+7 = 19 (also prime).

(What other numbers have this property?)

350^2 + 1 is prime (It is a twin prime with 350^2 + 3).

350^6 + 350^5 + 350^4 + 350^3 + 350^2 + 350^1 + 1 is prime.

The sum of the first 350 primes is prime.

There are 350 nonsquare rectangles on a 6 by 6 board.

350^2//350//1 is prime (// is the concatenation function. Thus 1225003501 is prime). (What's another number with this property?)

If 350 = n, n0123456789 is prime (Meaning 3500123456789 is prime). (What's the next number with the same property?)

###
December 17th (351st Day of the Non-Leap Year)

Dec 17th, 12/17.

1217 = 1271 - (7+2*1)*(7+1-2*1)

1217 = 1172 + (7*1+2*1)*(7*1-2*1)

1217 = 1127 + (7+2*1)*(7+2+1)

1217 = 1712 - 271 - 217 - (7*1)*(2-1)

1217 = (1+217) + (12+17) + (121+7) + (71+21) + (712+1) + (12+(7-2)^(1*2*1))

1217 = 1*217 + 12*17 + 121*7 - (1+2)*17

1217 = 71*21 + 712*1 - 1127 + 12^((7-1)/2-1) - (7-2-1-1)

1217 = (12+17)*(7*(7+1-2*1)) - (17-12)/(7*1-2*1)

1217 = (1+217)*(7-2) + 127

1217 = (121+7)*(1*2+1+7) - 7*(7*1+2*1)

1217 = (121-7)*(1+2+1+7) - (21+2^(7-2-1))

1217 = (1+2+1+7)*((1+2)*(12+(7-2)^((7-1)/2-1))) - (7-2-1)

1217 = (1*2*1*7)*(71+(2^(7-1-2))) - (17-12)/(7-2)

1217 is the 199th prime (199--prime).

1217 is prime.

1217^2 + 1217 + 1 is prime.

1217 is prime.

12+17 = 29 is prime.

1+2+1+7 = 11 is prime.

(What's the next prime with this property?)

351st Day of the Year.

351 = 315 + (5+1)^(3-1)

351 = 3*51 + 35*1 + 1*53 + 15*3 + 13*5

351 = (3+51) + (35+1) + (1+53) + (15+3) + (15+(5-3))^(5-3*1) - (5*(3-1))^(1*5-3)

351 = (3+5+1)*(51-(15-3))

351 = (3+51)*(13/(5-3))

351 = (35+1)*((31+(5+3))/(5-1))

351 = (35-1)*(5*(3-1)) + ((35-(5-3))/(5+1-3))

351 is the smallest number which, when raised to the sixth power, has 6 zeroes (351^6 has 6 zeroes).

351^1, 351^2, and 351^3 uses digits less than or equal to 5. (What's another number with this property?)

1+2+3+4+...+26 = 351.

351*(3+5+1) = 3159 (uses 3,5,1 and 3+5+1 as digits). (Are there other numbers with this property?)

351 is a divisor of 999999.

351^2 + 351 + 1 and 351^2 + 351 - 1 are twin primes. (What's the next number with this property?)

351 = 61 + 67 + 71 + 73 + 79 (5 cons primes).

###
December 18th (352nd Day of the Non-Leap Year)

Dec 18th, 12/18.

1218 = 1182 + (8-2)^(1+1)

1218 = 1812 - 281 - (8-1)^(2+1) + (12+18)

1218 = 1281 - (8+1)*(8-1)

1218 = 1821 - 812 + 281 - (1+2+1+8)*(18-12)

1218 = 1*218 + 12*18 + 121*8 - 182 - 8/(1+2+1)

1218 = 81*21 - 812*1 + 281 + 12*(8/(1*2*1))

1218 = (1+218) + (12+18) + (121+8) + (81+21) + (812+1) - (8*1-2-1)*(18-(1+2))

1218 = (12+18)*((8*1+2*1)*(8/(1*2*1))) + (1+2)*(1+8)

1218 = (121+8)*(18/(2*1)) + (81-(28-8/(1*2*1)))

1218 = (1+218)*(8-2-1*1) + 121 + (8/(1+2+1))

1218 = (1+2+1+8)*(121-(12+1*8)) + (18-12)

1218 = (1*2*1*8)*((18+(2-1))*(8/(1*2*1))) + (8/(1+2+1))

1218 + 1*2*1*8 = 1234 (ordered digits)

1218^23 + 1218^21 + 1218^19 + 1218^17 + 1218^15 + 1218^13 + 1218^11 + 1218^9 + 1218^7 + 1218^5 + 1218^3 + 1218^1 + 1 is prime. (What's another number that satisfies this property?)

There are 1218 6-digit emirps (primes that are prime when written backwards).

352nd Day of the Year.

352 = 325 + 3^(5-2)

352 = 253 + (3*(5-2))*(5+3*2)

352 = 235 + (3*5-2)*(3*(5-2))

352 = 3*52 + 35*2 + 2*53 + 25*3 - (3+52)

352 = (3+52) + (35+2) + (2+53) + (25+3) + 3*(52+(5+2))

352 = (5+2)^3 + (3*(5-2))

352 = (5*2+2^3)^(5-3) + (25+3)

352 = (3+5+2)*((5+2)*(3+2)) + (5+3-2)/3

352 = (3*5*2)*(3*5-(5-2)) - (5^2-(3-2))/(5-2)

352 = (35+2)*(3*(5-2)) + (23-(5-(3-2)))

352 = (3+52)*(3+5-2) + (25-3)

352 = (35-2)*(5*2+(3-2)) - (2*3+5)

352 = (25-3)*((2*(5-3))^(5-3))

There are 352 ways to place 9 nonattacking queens on a 9 by 9 board.

352^4 + 1 is prime. (What's another number like this?)

3+5+2 = 10.

3*5*2 = 30. (Multiple of 10)

(What other numbers satisfy this property?)

There are 352 primes between 50 and 50^2.

353^11 - 352^11 is prime. (What's the next number with the same property?)

If 352 = n, 2n3n5n7n11 is prime (Meaning, 235233525352735211 is prime). (What's the next number with this property?)

352 written in bases 17 through 36 contain at least one letter. (What's another number with this property?)

###
December 19th (353rd Day of the Non-Leap Year)

Dec 19th, 12/19.

1219 = 1192 + (12-9)^(9/(2+1))

1219 = 1129 + (92-1-1)

1219 = 1912 - 291 - 192 - 219 + (9*1*(2-1))

1219 = 1*219 + 12*19 + 121*9 - 291 - (19+(19-12))

1219 = 91*21 - 9*121 + 219 + 192 - (9-2)*(9/(2+1)-1)

1219 = (1+219) + (12+19) + (121+9) + (91+21) + (912+1) - 192 + (9-1-2-1)

1219 = (1+219)*((1*2+1*9)/(9/(2+1)-1)) + (9*(2-1))

1219 = (12+19)*(19+(1*2*(1+9))) + (9-1+2*1)

1219 = (121+9)*(9*(2-1)) + (9-2)^(12-1-9)

1219 = (121-9)*(1*2+1*9) - (1+2+1+9)

1219 = (1+2+1+9)*(91+2+1) - (9*1)/(2+1)

1219 = (1*2*1*9)*((19-2*1)*(12-(9-1))) - (9-1-2-1)

1219*9121 begins with a run of four 1s (11118499). (Do other numbers have this property?)

If 1219 = n, 101001000n is prime (Meaning, 1010010001219 is prime-- Note the initial string concatenates 10^1,10^2,10^3). (What's the next number that satisfies this property?)

If 1219 = n, 7nn7 is prime (Thus, 7121912197 is prime).

1219 written in bases 14 and 32 use the same digits (631 and 163, respectively).

1219 written in bases 16 and 17 use the same digits (4C3 and 43C, respectively).

(What's a different number with a similar property?)

353rd Day of the Year.

353 = 335 + (3+5*3)

353 = 3*53 + 35*3 + 33*5 - 53 - (35-(5*3-3))

353 = (3+53) + (35+3) + (33+5) + (3*5)^(5-3) - (5-3/3)

353 = (3*3-(5-3))^(3*(3+3-5)) + 5*(5-3)

353 = (3+5*3)^(5-3) + (33-(5-3/3))

353 = (3+5+3)*(35-3) + (3-5+3)

353 = (3*5*3)*((5-3)^3) - (5*3-(5-3)^3)

353 = (35+3)*(3^(5-3)) + (3+5+3)

353 = (3+53)*(3*(5-3)) + (5+3*(5-3/3))

353^2 = 124609 (1 < 2 < 4 < 6 < 9,exclude zeros).

Further 353^2 does not contain digits in 353.

(What's another number with both these properties?

353 is the 71st prime (71 is prime).

3+5+3 = 11 is prime.

The digits of 353 is prime.

(What's another prime with at least two of these properties? All three?)

353!!!!! + 1 is prime. (What's the next number with this property?)

353 is the smallest number such that its fourth power is the sum of four other fourth powers (353^4 = 30^4 + 120^4 + 272^4 + 315^4).

353 written in any base 2 through 10 uses digits no higher than 5. (What other numbers have this property?)

###
December 20th (354th Day of the Non-Leap Year)

Dec 20th, 12/20.

1220 = 1202 + (20-1*2)

1220 = 2012 - 1022 + 210 - (1-2)*20

1220 = 1*220 + 12*20 + 20*12 + 21*20 + (12-2)^(2+0)

1220 = (1+220) + (12+20) + (122+0) + (2+21) + (22+1) + (0+221) + (12*2)^(2+0) + 20/(12-1*2)

1220 = (12+20)*(20+(20-(2+0))) + 1*2*2

1220 = (1+220)*(1+2+2+0) + (1+2+2+0)! - (1+2+2+0)

1220 = (122+0)*(12-(2+0))

1220 = 12*220 - 1202 - (12/2)^(2+2-1) + (1-2)*(2+0)

1220 = (12*20)*(1+2+2+0) - (1-2)*20

1220 = (1+2+2+0)*(12^2 + 10^2)

1220 = (1*2*2)*(220 + (1+2+2+0)*(20-(1+2)))

Of all the primes less than 10^17, the furthest consecutive primes are 1220 apart.

If 1220 = n, 2n3n5n7 is prime (Meaning, 2122031220512207 is prime). (What's the next number with this property?)

1220 written in bases 11,12,15,16, and 21 is a palindrome (A0A, 858, 565, 4C4, and 2G2). (What's another number with 5 palindromes? 6 palindromes?)

354th Day of the Year.

354 = 345 + 3^(3-(5-4))

354 = 435 - (5+4)^(5-3)

354 = 3*54 + 35*4 + 4*(4^(5-3) - 3*(5-4))

354 = 4*53 + 45*3 + (3*4-5)

354 = (3+54) + (35+4) + (4+53) + (45+3) + (5-4)^3 + (5*(4-3))^3 + (3*(5-4))^3

354 = (3+5+4)*(34-5) + 54/(3^(5-3))

354 = (3*5*4)*(4+5-3) - 54/(5*4-(3*5-4))

354 = (35+4)*(45/(5*(4-3))) + 3*(5-4)

354 = (35-4)*(3*5-4) + (5*4-(4+3))

354 = 2*3*59 (2 < 3 < 59 and 2 < 3 < 5 < 9). (What's another number with this property?)

354 = 1^4 + 2^4 + 3^4 + 4^4

There are 354 12-digit left-truncatable primes.

Of all the primes less than 10^10, the maximum difference between two consecutive primes is 354.

###
December 21st (355th Day of the Non-Leap Year)

Dec 21st, 12/21.

1221 = (12-1)^(2+1) - 11*(12-2*1)

1221 = (12+(1*2+21))^(2*1) - (1*2*2*1)

1221 = 1122 + 11*(21-12)

1221 = 1212 + (1+2)^(2*1)

1221 = (1*221) + (12*21) + (122*1) + (21*21) + (12*12) + (21+(21-(2-1)))

1221 = (1+221) + (12+21) + (122+1) + ((1+2)*(2^(2+1)-1))^(2*1) + 2*221 - (2^(2+1))*(2+2+1)

1221 = (1+2+2+1)*(211-2^(2+1)) + 12/(2+1+1)

1221 = (1*2*2*1)*(221+12*(2^(2+1)-1)) + (1+2)/(2+1)

1221 = (12+21)*(21+2^(2+1+1))

1221 = (1+22+1)*(21+(2+1)*(12-2*1)) - (12/(2+2))

1221 = (1+221)*((12-1)/2)

1221 = (122+1)*(12-2*1) - (1+2)*(2+1)

12 and 21 are reverse numbers. So are 12^2 (144) and 21^2 (441).

1221^23 + 1221^21 + 1221^19 + 1221^17 + 1221^15 + 1221^13 + 1221^11 + 1221^9 + 1221^7 + 1221^5 + 1221^3 + 1221^1 + 1 is prime. (What's the next number with this same property?)

1221 is a divisor of 111111. (What's the next highest number that is a divisor of 1111111 (7 ones, not 6)?)

If 1221 = n, n^2 + n + 1, n^3 + n + 1, and n^4 + n + 1 are prime. (What's another number with this property?)

355th Day of the Year.

355 = 535 - 3*5*5 - 5*(5+(5-3)^(3+5/5))

355 = 35*5 + 3*55 + 5*((5-3)+(5/5))

355 = 5*53 + 55*3 - 3*5*5

355 = (3+55) + (35+5) + (5+53) + (55+3) + (3*(5-5/5))^(5-3) - (5-(5-3))

355 = (3+55)*(3*(3-5/5)) + (5+5-3)

355 = (35+5)*(3^(3-5/5)) - 35/(5+5-3)

355 = (3+5+5)*(35-(5-3)^3) + (3+5/5)

355 = (3*5*5)*(35/(5+5-3)) - (35-3*5)

355 = (35-5)*((3+5)+(3+5/5)) - 55/(3*5-(3+5/5))

355 = (55-3)*(5+5-3) - 3^(3-5/5)

355 = 5*71

5+7+1 = 3+5+5, thus 355 is the last Smith Number of the year. (What's the first nontrivial Smith Number of the year?)

355^10 + 355^9 + 355^8 + 355^7 + 355^6 + 355^5 + 355^4 + 355^3 + 355^2 + 355^1 + 1 is prime. (What's the next number with this property?)

There are 71 primes less than 355 (71 also divides 355).

355 is the numerator of the best simplified fraction approximating pi to 6 decimal places (355/113). To approximate pi to 7 decimal places, the denominator must be over 25,000.

###
December 22nd (356th Day of the Non-Leap Year)

Dec 22nd, 12/22.

1222 = 1*222 + 12*22 + 122*2 + 2*221 + (22+(2+2)*(2^(2+1)-(2/2)))

1222 = 22*21 + 212 + 221 + 222 + 122 - (22-(1+2+2))

1222 = (1+222) + (12+22) + (122+2) + (2+221) + (22+21) + (222+1) + 12*22 + (1*2*2*2)*(12-2/2)

1222 = (1+222)*(1+2+2) + 122 - (22-(2*2*2-1))

1222 = (12+22)*((2+2+2)^(1*2)) - (1*2*(2/2))

1222 = (122+2)*(22-12) - (12+(2+2+2))

1222 = (122-2)*(22-12) + (12-2/2)*(2*(2/2))

1222 = (1+2+2+2)*(122+2*(22+2*2)) + 12/(2+2/2)

1222 = (1*2*2*2)*(122+(22+(1+2*2*2)))

1222 = 2*13*47 (each prime factors uses different digits). (What's other number with this property?)

122201, 122203, 122207 and 122209 are all primes. (What's the next number with the same property?)

1222^13 + 1222^11 + 1222^9 + 1222^7 + 1222^5 + 1222^3 + 1222^1 + 1 is prime. (What's the next number with this property?)

356th Day of the Year.

356 = 365 - 36/(3+6-5)

356 = 536 - 5*36

356 = (3*56) + (35*6) - (5-3)*(6+5)

356 = (6*53) + (65*3) - 3*5*6 - (65+(5-3))

356 = (3+56) + (35+6) + (6+53) + (65+3) + 3*5*6 + 3*(6*(5-3))

356 = (3+56)*(6*(6-(6-3)-(6/3))) + (5-6/(5-3))

356 = (35+6)*(3^(5-3)) - (3*6-5)

356 = (65+3)*(5*((6/3)/(5-3))) + (5-3)^(6-(5-3))

356 = (65-3)*(5+(6-3)-(5-3)) - (3*5+(6-5))

356 = (35-6)*(36/(3*(6-5))) + (5-3)^(6-3)

356 = (3+5+6)*(5^(6/3)) + ((5+3)-(5-3))

356 = (3*5*6)*(3+6-5) - (3+6-5)

356 = 2*2*89 (prime factors do not use 3,5 or 6). (What's the next nonprime and nonsemiprime with this property?)

356 is the last Self number of the year (not of the form N + DigitSum(N) for some N).

[356,356*2] contains more primes than [356^2,357^2] (This is the last number of the year with this property).

There are 356 composites less than 21^2 (and including 21^2).

###
December 23rd (357th Day of the Non-Leap Year)

Dec 23rd, 12/23.

1223 = 1232 - 3^(1+2/2)

1223 = 1322 - (12-(1+2))*(23-12)

1223 = 1*223 + 12*23 + 122*3 + 322*1 + (3*2)^(2*1)

1223 = 32*21 - 3*221 - (12+(2+2)^(3-1))*(12/(3*(2/2)))

1223 = (1+223) + (12+23) + (122+3) + (3+221) + (32+21) + (322+1) + 232 + (2^3-1^2)

1223 = (1+223)*(3*2-(2-1)) + (123-(3+2)*(2*2*1))

1223 = (12+23)*(32+2+1) - (3-2)*(2*1)

1223 = (122+3)*(1*2+2^3)

1223 = (1+2+2+3)*(123+(32-2*1)) - (3-2)*(2-1)

1223 = (1*2*2*3)*(123-21) - (12/(1*2*2*3))

1223 = (122-3)*(23-(12+(3-2)) + (12+(2-3))*(3*(2-(2-1)))

1223 = (32+21)*(23*(2-1)) + (3*(2/2)+1)

1223 is prime. The next prime is 1229.

12231229 is also prime.

(What other consecutive primes also have this property?)

Saying 1223 digit by digit ("one two two three") spells out the number 233. 233 is also prime.

Additionally, the number describing 1223 (112213) is prime. Again, the number describing 112213 is prime. This continues two more times such that all the numbers listed below describe the previous number and are prime:

233, 1223, 112213, 21221113, 1211223113, 11122122132113.

1223 and 3221 are prime.

357th Day of the Year.

357 = 375 - (3*(5+(5-(7-3))))

357 = 3*57 + 35*7 - (57+(7-5))

357 = 7*53 + 75*3 - 375 + 3*5*7 + (35-(7-3))

357 = (3+57) + (35+7) + (75+3) + (7+53) + 3*5*7 + (3*((7-5)+(5-3)))

357 = (3+57)*(3*(7-5)) + (5*3-(7+5))

357 = (35+7)*((7*3-(7-3))/(5-3))

357 = (35-7)*((7*5+(5-3)^(7-3))/(7-3))

357 = (75+3)*((3^(7-5))/(5-3)) + 3*(7-5)

357 = (3+5+7)*(3*5+(3^(7-5))) - 3/(3-(7-5))

357 = (3*5*7)*(5-(7-5)/(5-3)) - 7*(3^(7-5))

357*753 = 268821 (does not contain 3,5 or 7). (Do other numbers have this property?)

There are 357 primes less than 2400 (Also the same number of primes are below 49^2).

357^2 + 4 is prime. (What's another number like this?)

357^43 + 357^41 + 357^39 + 357^37 + 357^35 + 357^33 + 357^31 + 357^29 + 357^27 + 357^25 + 357^23 + 357^21 + 357^19 + 357^17 + 357^15 + 357^13 + 357^11 + 357^9 + 357^7 + 357^5 + 357^3 + 357^1 + 1 is prime. (What's the next number with this property?)

###
December 24th (358th Day of the Non-Leap Year)

Dec 24th, 12/24.

1224 = 1242 - 2*(2*4+1)

1224 = 1422 - (22*(12-(4+2)/(2*1)))

1224 = ((1*2)*224) + (12*24) + (122*4)

1224 = 42*21 + 242 + (12-4/2)^(4-(2/2+1))

1224 = 4*221 + 422*1 - 2*41

1224 = (1+224) + (12+24) + (122+4) + (4+221) + (42+21) + (422+1) + (124+2)

1224 = (1+224)*((4+2)-(2-1)) + (1+2+2+4)*(22/(4-2))

1224 = (12+24)*(24+(12-(4/2)))

1224 = (122+4)*(1*2+2*4) - (4+2)^(2*1)

1224 = (122-4)*(24-(12+4/2)) + 4*(22/(4-2*1))

1224 = (1+2+2+4)*(124+(1+2)*4)

1224 = (1*2*2*4)*((122+(22+(1+2+2+4)))/(4-2))

1224 = 2*2*2*3*3*17

2+2+2+3+3+17 = 29 (also prime).

1224 = 2*2*2*3*3*17

1224*2223317 = 2721340008 (contains a run of 3 zeros). (1224 and 2223317 do not contain any zeros. Do other numbers have the same property?)

There are 1224 twin prime pairs less than 10^5.

1224^8 + 1 is prime. (What's the next number with the same property?)

1224 has 24 divisors (24 is in the string 1224). (What's the next number with this property?)

1224 = 35^2 - 1. 35 = 6^2 - 1 (both are of the form n^2 - 1).

358th Day of the Year.

358 = 385 - 3^(8-5)

358 = (3*58) + (35*8) - 3*5*8 + (35-(3+8))

358 = 8*53 + 85*3 - 385 + 8^(5-3)

358 = (3+58) + (35+8) + (8+53) + (85+3) + 3*5*8 - ((8+5)+(5-3))

358 = (35+8)*(3+5) + (3*5-(8/(3+5)))

358 = (3+58)*(8-(5-3)) - (3+5)

358 = (35-8)*(5+8) + (3*5-8)

358 = (85+3)*(3+8/(5+3)) + (8+3-5)

358 = (85-3)*(8/(5-3)) + 5*(8-(5-3))

358 = (3+5+8)*(35-(5+8)) + (3*8)/(5-(8/(5+3)))

358 = (3*5*8)*(8-5) - (8-(8+3-5))

358 = 2*179

2+179 = 181 (also prime).

2+1+7+9 = 19 (also prime).

(Do other numbers have this property?)

358 = 2*179 (prime factors don't include 3,5, or 8). (What's the next number with this property? Does a number use all numbers 0 through 9?)

The sum of the first 358 primes is prime. (What's the next number with this property?)

The digits of 358 are consecutive numbers in the Fibonacci sequence.

358 = 47 + 53 + 59 + 61 + 67 + 71 (6 cons primes).

###
December 25th (359th Day of the Non-Leap Year)

Dec 25th, 12/25.

1225 = (25+5*(1+2/2))^(5-2-2+1)

1225 = 1252 - (1*2+25)

1225 = 1522 - 252 - 5*(5+2+2*1)

1225 = (1*225) + (12*25) + (122*5) + 5*(22-(5-1))

1225 = (52*21) + 152 - (12+2+5)

1225 = (5*221) + (1+2+2)!

1225 = (522*1) + (12*52) + (52+(5-2)^(2+1))

1225 = (1+225) + (12+25) + (122+5) + (5+221) + (52+21) + (522+1) + (1+2+2*5)

1225 = (1+2+2+5)*((252-(5+2))/(5-2-2+1))

1225 = (1*2*2*5)*((225+(1*2*2*5))/(5-(2/2)))

1225 = (12+25)*((5-2)*(22/(5-2-2+1))) + (12/(5-2))

1225 = (1+225)*(5*(2-2+1)) + 5*(12+2+5)

1225 = (122+5)*(5*2-(2-1)) + (52+(5*2*(2+1)))

1225 = (122-5)*(5*2*(2-1)) + 22*((5*1)/(2*(2-1)))

25 = 5^2.

225 = 15^2.

1225 = 35^2.

(What's the next number or set of numbers that needs to precede 1225 to make a square?)

1+2+2+5 = 10.

1*2*2*5 = 20 (Multiple of 10).

(What's another number with the same property?)

1+2+3+...+48+49 = 1225.

10^1225 + 49 is prime (This number is 1000...00049 with 1223 zeros between 1 and 49).

1225 = 1^3 - 2^3 + 3^3 - 4^3 + 5^3 - 6^3 + 7^3 - 8^3 + 9^3 - 10^3 + 11^3 - 12^3 + 13^3.

1225 written in bases 17, 32, 33, 34, and 35 are also squares (441, 169, 144, 121, and 100, respectively).

359th Day of the Year.

359 = 395 - (9-3)^(5-3)

359 = (3*59) + (35*9) - 3*5*9 + (5-9/3)

359 = (9*53) + (95*3) - 395 - (5+9/3)

359 = (3+59) + (35+9) + (9+53) + (95+3) + 93*(9-5-3)

359 = (3+5+9)*(3*(9-(5-3))) + (5-9/3)

359 = (3*5*9)*(39/(3*5-(5-9/3))) - (39+(9-5+3))

359 = (3+59)*(3*5-9) - (5+(5+9/3))

359 = (35+9)*(5+9/3) + (9+5)/(5-3)

359 = (35-9)*(5+9) - (3+(9-5)/(5-3))

359 = (95+3)*(9-5) - (3*(9+5-3))

359 = (95-3)*(3*5-(9+5-3)) - 3^((9-5)/(5-3))

359^2 = 128881 (does not contain 3, 5, or 9). (What's another prime with the same property?)

359 and 953 are primes.

359 is prime. 3+5+9 = 17 is prime.

Inserting a 3 between any adjacent digits in 359 (or in the beginning/end) result in a prime (i.e. '3'359, 3'3'59, 35'3'9, and 359'3' are all prime, where ' ' represents the new 3 being added). (What's a number with this property but the number does not initially contain a 3?)

359!!! + 1 is prime. (What's the next prime with this property?)

Of all the 4-digit primes, 359 twos appear in their decimal representation.

359^128 + 360^128 is prime.

359 is of the form 8n-1 (n=45).

With n = 45.... 4n-1, 8n-1, 16n-1, 32n-1, and 64n-1 are all primes (359 is the first number with this property). (What's the next number with this property?)

###
December 26th (360th Day of the Non-Leap Year)

Dec 26th, 12/26.

1226 = 1262 - (2*(1+2))^(6/2-1)

1226 = 1622 - 262 - 162 + (1*2+26)

1226 = (1*226) + (12*26) + (122*6) - (6-2)*(22/(6/2-1))

1226 = (6*221) - (1*2+2+6)^(6-2-2*1)

1226 = 62*21 - (26+2*(6-1)^2)

1226 = (1+226) + (12+26) + (122+6) + (6+221) + (62+21) + (622+1) - (12-(6/2-1))^(6/2-1)

1226 = (1+2+2+6)*(126-(12+6/2)) + (12-(6+2+1-2))

1226 = (12+26)*(2^(6-1)) + (26-(6-2)^(1*2))

1226 = (122+6)*(12-6/2) + (62+12)

1226 = (1+226)*((6/2)+(2/1)) + (61+(6*(2+2+1)))

1226 = (122-6)*(1+2+2+6) - ((6-1)^2)*(6/2-1)

1226 = (62+21)*(12+6/2) - (12+(1+6))

1226 = (1*2*2*6)*(62-(22/(6/2-1))) + (6-2-2*1)

360th Day of the Year.

360 = 306 + 6*(6+0+3)

360 = (3*60) + (6*30)

360 = (3*6)*(60/3)

360 = (3+60) + (36+0) + (0+63) + (30+6) + 3*60 - 3*6

360 = (3+6+0)*(60-(60/3))

360 = (3+60)*6 - 3*(6+0)

360 = (30+6)*(60/3 - (3+6+3^0))

360 has a prime factorization of three 2s, two 3s, and one 5 (Note: three - two = one and 2 + 3 = 5).

360 = 2*2*2*3*3*5

360*222335 = 80040600 (only contains 3 nonzero digits, all even). (Do other numbers have this property?)

360^32 + 1 is prime. (What's the next number (ending in a 0) that has this property?)

There are 360 possible rook moves on a 6 by 6 board.

The 359th, 360th, and 361st digits of pi make the number 360.

The same number of primes are in [360,2*360] and [360^2,361^2].

There are 360 degrees in a circle.

360 is the smallest number divisible by all numbers 1-10 (except 7).

There are 72 primes below 360 (72 is also a divisor of 360).

###
December 27th (361st Day of the Non-Leap Year)

Dec 27th, 12/27.

1227 = 1272 - (7+2)*(7-2)

1227 = 1722 - 272 - 212 - (1*2+2+7)

1227 = (1*227) + (12*27) + (122*7) - 172 - (12-(7-1))

1227 = (7*221) - 272 - 12*(7-2-2+1)

1227 = (72*21) - (7*2-(2-1)) - 272

1227 = (1+227) + (12+27) + (122+7) + (7+221) + (72+21) + (722+1) - 272 + (72-(7*2-1))

1227 = (1+2+2+7)*(122-(12+(2+7))) + (7*2+(2-1))

1227 = (1*2*2*7)*(22*(7-2-2-1)) - (7-2)*(2-1)

1227 = (12+27)*(27+(7-2-2+1)) + (12+(7-2-(1-2)))

1227 = (122+7)*((2-1)*(7+2)) + 22*(7-2-2*1)

1227 = (122-7)*(22/(7-2-2-1)) - ((7-2-2-1)+(7-1)^(7*2-12))

1227 = (72+21)*((1+2)*2+7) + (27-(2+7))

361st Day of the Year.

361 = 316 + (6+3)*(3+6/(3*1))

361 = 163 + 136 + (63-1)

361 = (3*61) + (36*1) + (1*63) + (16*3) + 31

361 = (3+61) + (36+1) + (1+63) + (16+3) + 163 + (6+1)*(3-1)

361 = (3+6+1)*(36*1) + (6/3-1)

361 = (3*6*1)*((6-1)*(3+1)) - (1-6/3)

361 = (6+1)^3 + 3*6*1

361 = (36+1)*(3+6+1) - (3+6*1)

361 = (3+61)*(3+(6/(3-1)) - (3*6*1+(6/(3-1)+6/3))

361 = (36-1)*(3+6+1) + (3+(6+(3-1)))

361 = (16+3)^(6-3-1)

361 = 19^2. Removing the 1 still results in a square (6^2).

Further, 361 is a square that is a concatenation of two squares (6^2 and 1^2).

1 +- 2 +- 3 +- 4 +- 5 +- 6 +- 7 +- 8 +- 9 +- 10 +- 11 +- 12 +- 13 +- 14 +- 15 = 0 has 361 different solutions (depending on +/-).

361 written in bases 7, 9, 16, 17, 18, and 19 are squares (1024, 441, 169, 144, 121, and 100, respectively).

###
December 28th (362nd Day of the Non-Leap Year)

Dec 28th, 12/28.

1228 = 1282 - (8-2)*(8+1)

1228 = 1822 - 282 - 218 - (82+12)

1228 = (1*228) + (12*28) + (122*8) - 282 - (1*2+28)

1228 = 8*221 - 282 - 228 - (1*2+28)

1228 = 82*21 - 228 - 282 + (8*2*(2-1))

1228 = (1+228) + (12+28) + (122+8) + (8+221) + (82+21) + (822+1) - 282 - (8+2+2-1)*((8/2)*(2-1))

1228 = (1+2+2+8)*(82+12) + (8-2)*(2-1)

1228 = (1*2*2*8)*(28+(2+8)) + (1*2+(2+8))

1228 = (12+28)*((1+2)+28) - (28-(2*8))

1228 = (1+228)*(8/2+(2-1)) + (82+(2-1))

1228 = (122+8)*((8/2-1)^2) + (28+(2*(12+(8/2-1))))

362nd Day of the Year.

362 = 326 + (3*2)^(6/3)

362 = 236 + 23*6 - (3*2+6)

362 = 263 + (3^2)*(3+6+2)

362 = (3*62) + (36*2) + (62+(3+6-2)*(3*2))

362 = (2*63) + (26*3) + 32*6 - (36-2)

362 = (3+62) + (36+2) + (2+63) + (26+3) + 3*62 - (3*(6+3-2))

362 = (3+6+2)*((3+6+2)*(6/2)) - (6/3)/2

362 = (3*6*2)*(26-(6-2)^(6/3)) + (6/3)

362 = (36+2)*(6*2-3) + (6+2-3)*(6/3+2)

362 = (36-2)*(3+6+2) - (6+(3*2))

362 = (3+62)*(6+2-3) + (36+(3-2))

362 = (26+3)*(36/(6-3)) + (2^3+6)

362 = (26-3)*((6-2)^(6/3)) - (2*(6-3))

3*6*2 = 36 (36 is in 362). (What's the next number like this?)

362^8 + 363^8 is prime. (What's the next number with this property?)

x^x^x^...^x (with 33 x's) has 362 different 3rd derivatives (depending on the parentheses) at x = 1.

362^2 + 363^2 + 364^2 is prime. (What's another number with the same property?)

###
December 29th (363rd Day of the Non-Leap Year)

Dec 29th, 12/29.

1229 = 1292 - (9-2)*((1+2)^2)

1229 = 1922 - 292 - 129 - 219 - (29+12*2)

1229 = 122 + 229 + 292 + 219 + 192 + 129 + (29 + (2*9+(1-2)))

1229 = 922 + 292 - (9*2-(2+1))

1229 = (1*229) + (12*29) + (122*9) - 292 - 129 + (9-2-2*1)^((9-2-2-1)/2)

1229 = (9*221) - 922 + 129 + (1+2)*(2+9)

1229 = 92*21 - 922 + 219

1229 = (1+229) + (12+29) + (122+9) - (9+221) + (92+21) + (922+1) + (9-2)*(2+1)

1229 = (12+29)*(29-(1-2)) - (12-(2+9))

1229 = (1+2+2+9)*(22*(9-2-2-1)) - (9-(1+2)*2)

1229 = (1*2*2*9)*(22+(1+2+9))

1229 = (1+229)*(9-2-2*1) + (92-(1*2+2+9))

1229 = (122+9)*(9*(2/2)) + (29+21)

1229 = (122-9)*((2-1)*(9+2)) - (1+2+2+9)

1229 is prime.

There are 1229 primes less than 10^4.

1229 is prime.

1229 - 1*2*2*9 = 1193 (also prime).

If 1229 = n, 101001000n is prime (Meaning, 1010010001229 is prime--note the initial concatenation: 10,100,1000). (What's the next number with the same property?)

1229^128 + 1230^128 is prime. (What's the next prime with this property?)

1229 and 9221 are prime.

1229 = 1234-5

363rd Day of the Year.

363 = 336 + 3*(3+6)

363 = 3*63 + 36*3 + (3+63)

363 = 33*6 + 3*6*3 + (36+3/3)*(6-3)

363 = (3+63) + (36+3) + (33+6) + 336 - 3*(36+3)

363 = (3+6+3)*(6*(6-3/3)) - (6+3)/3

363 = (3*6*3)*(6+3/3) - (6+3*3)

363 = (36+3)*(36/((6-3)+3/3)) + (3+6+3)

363 = (36-3)*(33/(6-3))

363 = (3+63)*((3+(6/3)^(6-3))/(6/3))

363 is the numerator of 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7.

363 is of the form p^2 + 2 (for prime p...here, p = 19).

Also, 363^2 + 2 is prime.

(What other numbers have this property?)

363!! + 32 is prime (This is the largest known number with this property).

There are 363 primes between [51,51^2].

363 = 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 (9 cons primes).

###
December 30th (364th Day of the Non-Leap Year)

Dec 30th, 12/30.

1230 = 1203 + 3^(2+1)

1230 = 1320 - (1+2)*30

1230 = 1302 - (1*2*3)*12

1230 = 1023 + 213 - 1*2*3

1230 = 1032 + 231 - 3*(13-2)

1230 = 1*230 + 32*10 + 12*30 + 3*21 + 32*1 + (12+3)^(3-2+1)

1230 = (1+230) + (12+30) + (123+0) + (0+321) + (3+21) + (32+1) + 312 + 12^(3-1)

1230 = (1+2+3+0)*(213-2^(1*3))

1230 = (1+230)*(1*2+3) + (3+2)*(12+3)

1230 = (12+30)*(30-(2-1)) + (23-(13-2))

1230 = (123+0)*(30/(1+2))

1230 = (30-12)*(32+(3*2)^(2*1)) + (1+2+3+0)

1230 = 2*3*5*41 (prime factors use all different digits, only digits 1 through 5). (What's another calendar day with this property?)

1230 = 16^2 + 17^2 + 18^2 + 19^2

If 1230 = n, 1 + n + n^3 + n^5 + n^7 + ...+ n^63 is prime. (What's the next number satisfying this property?)

The 1230th prime is the smallest prime with 3 zeros.

364th Day of the Year.

364 = 346 + (36/(6-4))

364 = 436 - 3*6*4

364 = 463 - 3*(34-(4-(6-3)))

364 = 3*64 + 36*4 + (34-6)

364 = 4*63 + 46*3 - (3+6+4)*(4-6/3)

364 = (3+64) + (36+4) + (4+63) + (46+3) + (3*4)^(6-4) - (6-3)*(4-(6-3))

364 = (3+6+4)*(34-6)

364 = (3*6*4)*(3+6-4) + (6/3)^(6-4)

364 = (3+64)*(6-(4-3)) + (6*3+(3+(6-4)^3))

364 = (36+4)*(3^(6-4)) + 64/(4^(6/3))

364 = (36-4)*(3+(6-4)^3) + (36/(6-3))

364 = (46+3)*(6+4-3) + 3*(6+4-3)

There are 364 gifts mentioned in "12 Days of Christmas".

364^4 + 1 is prime. (What other numbers are like this?)

There are 364 ways to place 3 nonattacking queens on a 3 by 10 board.

364 = 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0.

2^364 reversed is a prime. (What other numbers have this property?)

364^2 + 365^2 + 366^2 is prime. (What's the next number with the same property?)

If 364 = n, 2n3n5n7n11 is prime (Meaning, 236433645364736411 is prime). (What's another number with this property?)

If 364 = n, n0123456789 is prime (Thus, 3640123456789 is prime). (What's the next number with this property?)

364 written in bases 3, 9, 25, 27, 51 and 90 is a repdigit (111111, 444, EE, DD, 77 and 44, respectively). (Do other numbers have a similar property? Number with more repdigits?)

364 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 (12 cons primes).

###
December 31st (365th Day of the Non-Leap Year)

Dec 31st, 12/31.

1231 = 1321 - (31-(2-1))*(12/(3+1))

1231 = 1213 + (12+1*2*3*1)

1231 = 1312 - (3^(2+1+1))

1231 = 1132 + (3^2)*11

1231 = 1123 + (3^2)*((1+1)*2*3)

1231 = (1*231) + (12*31) + (123*1) + (13*21) + (132*1) + (12-(3-1))^(3-1)

1231 = (1+231) + (12+31) + (123+1) + (1+321) + (13+21) + (132+1) + (2^3-1)^(12/(3+1))

1231 = (1+2+3+1)*(123+(31+23-1)) - (1-2+3-1)

1231 = (1*2*3*1)*(213-(2^3)) + (1-2+3-1)

1231 = (12+31)*(31-1*2) - (1*2)^(3+1)

1231 = (13+21)*((2*3)^(1+1)) + (1+2+3+1)

1231 = (123+1)*(12-(3-1)) - (1+2^3)

1231 = (123-1)*(31-21) + (13-2*1)

1231 = (31-12)*(32+3*(12-1)) - (12/(3*1))

Each year, we end on a twin prime pair: 1229 and 1231.

If 1231 = n, n0123456789 is prime (Meaning, 12310123456789 is prime). (What's the next prime with this property?)

365th Day of the Year.

365 = 356 + 36/(6-(5-3))

365 = 536 - 3*6*5 - (3^(6-(5-3)))

365 = 563 - (6+5)*(6*3)

365 = 635 - 36*5 - 3*6*5

365 = 653 - 3*65 - 3*(36-5)

365 = (3*65) + (36*5) - (5*(6/3))

365 = (5*63) + (56*3) - 3*6*5 - (5+6/3)*(3+6-5)

365 = (3+65) + (36+5) + (5+63) + (56+3) + 3*6*5 + 3*(3*6-5)

365 = (3+6+5)*(2*(3*6-5)) - (6/3-(6-3))

365 = (3*6*5)*(3+6-5) + 5*((6-3)-6/3)

365 = (3+65)*(6/3+(6-3)) + 5^(6/3)

365 = (36+5)*(3^(3-(6-5))) - (3+6-5)

365 = (36-5)*(6*(5-3)) - (5+6/3)

The same number of primes are between [365,2*365] and [365^2,366^2].

There are 365 perfect squares less than 51^3.

There are 365 nonattacking knights on a 27 by 27 board.

10^2 + 11^2 + 12^2 = 13^2 + 14^2 = 365.

365 written in base 4 contains 1231 (for Dec 31). (Do other numbers have this property?)