July 1st (182nd Day of the Non-Leap Year)

Jul 1st, 7/1.

71 and 17 are both prime.

71^2 = 7!+1!

71^3 = 357911 (concatenation of odd numbers)

71^2-71-1 is prime.
71^2+71+1 is prime.

71^3-71-1 is prime.
71^3+71+1 is prime.

2^39 has a digit sum of 71. This isn't surpassed until 2^48 (73).

71^4-71-1 is prime.

3^71 +/- 20 are primes.

There are 71 different characters that can be used with a standard English keyboard (uppercase included).

71 divides 2+3+5+7+11+...+61+67+71.

The Earth is approximately 71% of Earth's surface.

2^71 does not contain a 9 (largest known prime exponent with this property).

Clint Eastwood is Inspector 71 in the movie Dirty Harry (which came out in 1971).

71π is the closest to a prime than any multiple of π below it.

182nd Day of the Year.

182 = 128 + (8+1)*(8-2)

182 = 218 - (18-12)^2

182 = 281 - (1+8+2)*(18/2)

182 = 812 - 81*2 - 8*12 - 12*(28+(8/2-1))

182 = 821 - 8*21 - 82*1 - 281 - (28-1)*(8/2)

182 = 1*82 + 18*2 + 1*8^2

182 = 2*81 + 28*1 - (18-(8+2))

182 = (1+82) + (18+2) + (82-(2+1))

182 = (2+81) + (28+1) + (8-1)*(8+2)

182 = (1+8+2)*(18-2) + (1*8-2)

182 = (1*8+2)*18 + (8/2-2/1)

182 = (1+8*2)*(1+8+2) - (8-(2+1))

182 = (18+2)*(18/2) + (18-2)/(8*(2-1))

182 = (18-2)*((1+2)*8/2) - (1*8+2)

182 = (1+82)*(8/2-2/1) + 1*8*2

182*(1+8+2) is a palindrome.

The 182nd prime + 1 is divisible by 182.

The 182nd and 183rd prime are twin primes.

There are 182 distinct values for the 5th derivative of x^x^x^x^x^x^x^x^x at x = 1.

182^4 + 183^4 is prime.

There are 182 squares < 32^3.

182^2 + 183^2 + 184^2 is prime.

182*181*180*179*...*3*2*1 / ((182nd prime)*(181st prime)*(180th prime)*...*(3rd prime)*(2nd prime)*(1st prime)) + 1 is prime.

In simplified notation, 182!/182# + 1 is prime.

182 can be expressed as the sum of 2 primes in exactly 6 different ways.

182^6+182+1 is prime.

182 = 3^5 - 3^4 + 3^3 - 3^2 + 3^1 - 1 = 182.

(1!)^2 + (2!)^2 + (3!)^2 + ... + (181!)^2 + (182!)^2 is prime.

1^1 + 8^2 + 2^3 is prime.

A Southern California pop punk band is named Blink-182.

July 2nd (183rd Day of the Non-Leap Year)

Jul 2nd, 7/2.

72 = 2*2*2*3*3
72*(2+2+2+3+3) = 864.
72*(7+2) = 648.

2^72 (22 digits) does not contain a 0.

72 = 2^3*3^2.

72^2+73^2 is prime.

72^6+72^5+72^4+72^3+72^2+72+1 is prime.

72^4+72+1 is prime.

72 = 13+17+19+23 (four consecutive primes).
72 = 5+7+11+13+17+19 (six consecutive primes).

72 degrees Fahrenheit in considered room temperature.

There are 72 average heartbeats per minute for a resting adult.

The human body is composed 72% of water.

The axis of the Earth moves one degree every 72 years compared to the stars.

72 is usually par score for an 18 hole golf course.

183rd Day of the Year.

183 = 138 + (1+8)*(8-3)

183 = 318 - (13-8)*(3*(1+8))

183 = 381 - 3*81 + (81-(8+1)*3)

183 = 831 - 8*31 - 83*1 - (13-8)*(18-1*8)^(3-1)

183 = 813 - 8*13 - 81*3 - 318 + (8-1)*(8-3)

183 = 1*83 + 18*3 + (3-1)*(3*8-1)

183 = 3*81 - 38*1 - (18+8/(3-1))

183 = (1+83) + (18+3) + 13*(8-(3-1))

183 = (3+81) + (38+1) + (1+3)*(18-3)

183 = (1+8+3)*(18-3) + 18/(8-(3-1))

183 = (1*8+3)*(3*(8-1)-(8-(3+1))) - 8/(3-1)

183 = (1+8*3)*(18/3+1) + (13-(8-3))

183 = (18+3)*(18/(3-1)) - (8-(3-1))

183 = (1+83)*(8/(3+1)) + (18-3)

183 has 361 pages in the OEIS (as of July 1st, 2014). And 3*61 = 183.

183^3-183-1 is prime.

183^4-183-1 is prime.

There are 183 primes < e^7.

184^11 - 183^11 is prime.

183*183! - 1 is prime.

183 is the smallest number where n//n+1 is a square (183184 = 428^2).

July 3rd (184th Day of the Non-Leap Year)

Jul 3rd, 7/3.

73^2 = 5329 (concatenation of two primes).

73 is the 21st (7*3) prime and 37 is the 12th prime (Note the reversal of digits).

Sheldon Cooper on the Big Bang Theory claims 73 is the best number.

73! + 1 is prime.

(19^73-3^73)/16 is prime (73 is the smallest number with this property).

73^4+73^3+73^2+73+1 is prime.

73^6+73^5+73^4+73^3+73^2+73+1 is prime.

73^2+4 is prime.

73 in the smallest integer with 12 letters in its English name.

73 is the only prime repunit in base 8 (111b8).

73 is the 21st prime. 73 is 1001001 base 2. 21 is 10101 base 2.

It took 73 seconds for the tragic explosion of the Challenger Space shuttle.

In international curling competitions, each side is given 73 minutes to complete all of its throws.

The Philadelphia 76ers lost in the 1972-73 season (9-73), it was the most losses in NBA history.

In the 1940 NFL championship game, the Bears beat the Redskins 73-0, the most points scored ever in an NFL game.

184th Day of the Year.

184 = 148 + 18*(8/4)

184 = 418 - 41*8 + (48-1)*(8/4*1)

184 = 481 - 4*81 + (8/4+1)^(4-1)

184 = 814 - 8*14 - 81*4 - (8/4)*(84+(1+8+4))

184 = 841 - 8*41 - 84*1 - (4+1)*(8-1)^(8/4)

184 = 1*84 + 18*4 + 4*(8-1)

184 = 4*81 - 48*1 - 4*(8*4-(8+1))

184 = (1+84) + (18+4) + (8-1)*(18-(8-1))

184 = (4+81) + (48+1) + (48+8/4)

184 = (1+8+4)*(18-4) + 1*8/4

184 = (18+4)*8 + (18-14)*(8/4)

184 = (1+84)*(8/4) + 14

184 = (1*8+4)*((4+1)*(4-1)) + (18-14)

184 = (1+8*4)*(1+4) + (14+(1+4))

184 = 2^3*23 (Note the pattern here).

184^2 + 1 is prime.

184^3-184-1 is prime.

184^3+3 and 184^3-3 are both prime.

184^4 + 185^4 is prime.

184^2 + 185^2 + 186^2 is prime.

July 4th (185th Day of the Non-Leap Year)

Jul 4th, 7/4.

74^2+1 is prime (and a twin prime, as well).

74^4+1 is prime.

74^16+1 is prime.

74^3+74+1 is prime.

74^4+74+1 is prime.

74^4+74^3+74^2+74+1 is prime.

2^74 (23 digits) does not contain the digit 2.

A hurricane or typhoon is a system with sustained winds of at least 74 mph.

185th Day of the Year.

185 = 158 + (8+1)*(8-5)

185 = 815 - 81*5 - 15^(8-(5+1))

185 = 851 - 8*51 - (1+5)*(58-15)

185 = 581 - 5*81 + 18/(8-(5+1))

185 = 518 - 5*18 - (8-5)^(15/(8-5))

185 = 1*85 + 18*5 + (18-1*8)

185 = 5*81 - 58*1 - (8-5-1)*(8-5)^(5-1)

185 = (1+85) + (18+5) + (5-1)*(15+(5-1))

185 = (5+81) + (58+1) + 1*5*8

185 = (1+8+5)*(1*8+5) + (1*8-5)

185 = (1+8*5)*5 - 5*(8/(8-5-1))

185 = (18+5)*8 + (15-(8-5-1)*(8-1))

185^5 is the next power of 185 to have an 8.

There is no prime between 1850 and 1859.

185^2 + 186^2 is prime.

185^3-185-1 is prime.

185^16 + 186^16 is prime.

185 is a semiprime and 185^8+1 is a semiprime.

185 = 6^3-6^2+6^1-6^0.

1000...0003000...0001 (185 digits) is prime.

The decimal expansion of the first 185 digits of e = 2.718281828... is prime.

July 5th (186th Day of the Non-Leap Year)

Jul 5th, 7/5.

75^2+75+1 is prime.

75^16+76^16 is prime.

75!! + 2 is prime.

75*2^75 + 1 is prime.

2^75+75 is prime.

The oldest a Canadian senator can be is 75.

186th Day of the Year.

186 = 168 + 6*(8-(6-1))

186 = 816 - 81*6 - (18-6)^(1*8-6)

186 = 861 - 8*61 - (18-(8-1))*(16+(8-6-1))

186 = 618 - 6*18 - 18^(8-6*1)

186 = 681 - 68*1 - 61*(6+1)

186 = 1*86 + 18*6 - 16/(8-6)

186 = 6*81 - 68*1 - 8*(6*(8-1)-(8+6-1))

186 = (1+86) + (18+6) + (8-(6-1))*(6-1)^(8-6)

186 = (6+81) + (68+1) + 6*(6-1)

186 = (1+8+6)*(18-6) + 18/(8-(6-1))

186 = (1*8+6)*(18-(6-1)) + (16-(18-6))

186 = (1+8*6)*(8/(8-6)) - (6-1)*(8-6)

186 = (18+6)*8 - (18-6*(8-6))

186*681 ends in 4 6's.

186^3+186+1 is prime.

186^3-186-1 is prime.

186^6 + 186^5 + 186^4 + 186^3 + 186^2 + 186 +1 is prime.

18600 - 18699 is the smallest interval of 100 with exactly 6 primes.

186^4+186+1 is prime.

186^8 + 187^8 is prime.

186^4 + 187^4 is prime.

There are 186 odd days in a non-leap year.

July 6th (187th Day of the Non-Leap Year)

Jul 6th, 7/6.

2^76 (23 digits) does not contain a 0.

(76!)^2+1 is prime (76 is the largest number with this property).

76^n ends in 76 for any n > 0.

17th prime + 17 = 59 + 17 = 76.

76^16+1 is prime.

76^3-76-1 is prime.

There are 76 ways to place 3 nonattacking queens on a 3 by 7 board.

There are 76 numbers <= 1000 with 5 prime factors.

76^8 + 77^8 is prime.

There are 76 primes between 14^3 and 15^3.

187th Day of the Year.

187 = 178 + (17-8)

187 = 781 - 7*81 - (18+(1+8))

187 = 718 - 71*8 + ((8*7-1)-18)

187 = 871 - 8*71 - (8-(7-1))*(78-(18+(8-(7-1))))

187 = 817 - 81*7 - (8+1)*7

187 = 1*87 + 18*7 - (18+1*8)

187 = 7*81 - 78*1 - 178 - (8-(7-1))*(71-(8+1))

187 = (1+87) + (18+7) + (78-8/(8-(7-1)))

187 = (7+81) + (78+1) + (1*8*7-(7-1)^(8-(7-1)))

187 = (1+8+7)*(18-(7-1)) - (17-(7-1)*(8-(7-1)))

187 = (1+87)*(8-(7-1)) + (18-7)

187 = (18+7)*(8-1) + (18-(7-1))

187 = (18-7)*17

187 = (1*8+7)*(18-(7-1)) + (78-71)

187 = (1+8*7)*(18/(7-1)) + (18-(8-(7-1)))

187^2 and 187^3 don't contain a 1, 7, or 8.

187^5 is the first power of 187 > 1 that contains a 7.

187*(1+8+7) is a palindrome.

1871, 1873, 1877, and 1879 are all prime.

187^2+188^2 is prime.

187^4+188^4 is prime.

187^4-187-1 is prime.

In a room of 187 people, there is a 50% chance that 4 people have the same birthday.

"187" is the police term for a murder or homicide; many gangs use it as well, in slang.

The 187th prime with 1117 and 11*17 = 187.

187^(1*8*7)+(1+8+7) is prime. Only one other number with nonzero digits has this property.

July 7th (188th Day of the Non-Leap Year)

Jul 7th, 7/7.

77!+1 is prime.

77^2-77-1 is prime.

77^2+77+1 is prime.

77^4+77^3+77^2+77+1 is prime.

2^77 (24 digits) does not contain a '0'.

77= 3^4-3-1 (3 is prime) and 77^4-77-1 is prime.

77^4+77+1 is prime.

There are 77 inequivalent ways to cut a 5 by 5 square into squares with integer sides.

77 is a palindrome and the 77th triangular number is a palindrome.

There are more primes between [77,2*77] than [77,77^2].

There are 77 numbers less than 4^5 with 5 not necessarily distinct prime factors.

2^77 reversed is prime.

77 is the smallest positive integer requiring 5 syllables.

188th Day of the Year.

188 = 818 - 81*8 + 18*(1*8/8)

188 = 881 - 8*81 - (1+8)*((18-8)/(1+8/8))

188 = 18*8 + 1*88 - (88/(1+8/8))

188 = (1+88) + (18+8) + (81-8)

188 = (1+8+8)*(18-(8-1)) + 1*8/8

188 = (18+8)*(8-1) + (8-(1+8/8))

188 = (18-8)*18 + (18-(8+1+8/8))

188 = (1*8+8)*(8+8/(1+8/8)) - (18-(8-1)*(1+8/8))

188 = (1+8*8)*(18/(8-(1+8/8))) - (88-81)

188^2 and 188^3 don't contain a 1 or 8.

188^16+1 is prime.

188^3-188-1 is prime.

188^5-188-1 is prime.

18800 to 18899 is the smallest interval of 100 with only 5 primes.

There are 188 primes between 24^3 and 25^3.

188^2 + 188 +/- 1 are twin primes.

There are more primes between 188 and 2*188 than 188^2 and 189^2.

188 is the largest number that can be expressed as the sum of two primes in exactly 5 different ways.

100^188+10^188-1 is prime.

1^1+8^2+8^3 and 1^3+8^2+8^1 are both prime.

\$188 is the most money you can have using only one US bill each (\$100, \$50, \$20, \$10, \$5, \$2, \$1).

July 8th (189th Day of the Non-Leap Year)

Jul 8th, 7/8.

78 = 1+2+3+4+5+6+7+8+9+10+11+12

56 = 7*8 (Note: 5,6,7,8 are consecutive).

78^2+78+1 is prime.

There are 78 primes less than 400.

There are 78 different values for 2^2^2^2^2^2^2^2^2 (depending on the placement of parentheses).

There are 78 ways to place 2 nonattacking kings on a 4 by 4 board.

78^8+79^8 is prime.

The digit 6 appears 78 times in the first 1000 Catalan numbers.

The digit 1 appears 78 times in the first 1000 prime numbers.

2*78! + 1 is prime.

9*10^78+1 is prime (That is, 9000...0001 is prime with 77 zeros).

There are 78 total gifts in the song Twelve Days of Christmas.

189th Day of the Year.

189 = 198 - (18-9)

189 = 819 - 81*9 + 9*(18-(8-1))

189 = 891 - 8*91 + (18+1*8)

189 = 918 - 91*8 - (9-8*1)

189 = 981 - 9*81 - 9*(8-1)

189 = 1*89 + 18*9 - (9*(8-1)-1)

189 = 9*81 - 98*1 - 198 - (19-(8-1))^(9-(8-1)) - (1+9)^(18/9)

189 = (1+89) + (18+9) + 1*8*9

189 = (9+81) + (98+1)

189 = (1+8+9)*(1+9) + 18/(9-(8-1))

189 = (18+9)*(8-1)

189 = (18-9)*(19+(9-(8-1)))

189 = (1+89)*(9-(8-1)) + (81-9*8*1)

189^2+189+1 is prime.

189^2 + 190^2 is prime.

189^4 + 190^4 isp rime.

189^3-189-1 is prime.

189^4-189-1 is prime.

2^189-189^2 is prime.

There are 189 zeros in the numbers 1 to 999.

The product of 4 distinct primes, all with a different last digit will always end in 189.

July 9th (190th Day of the Non-Leap Year)

Jul 9th, 7/9.

79^2+80^2 is prime.

79^4+79^3+79^2+79+1 is prime.

79^4-79-1 is prime.

There are 79 ways to place 4 nonattacking kings on a 4 by 4 board.

2^79 is the smallest power of 2 greater than Avogadro's Numbers (6.022e23).

79 = 2^7 - 7^2 (Note: 2+7 = 9).

The square root of 79 begins with four 8's.

In Illinois, USA, riding in the left hand lane for over half a mile is a \$79 fine.

1 = one --> 15+14+5 = 34

2 = two --> 20+23+15 = 58

...

Numbers 1-79 when written out in English and added up the letters, the letters are more than the number itself. It is not until 80 that the letters are less than 80.

79 = 11 + 31 + 37

97 = 11 + 13 + 73

190th Day of the Year.

190 = 109 + 9^(10-(9-1))

190 = 19^(1^9+9^0) - 9*19

190 = 910 + 9*10 - 91*9 + (1*9+0)

190 = 901 - 19*11 - 90*1 - 109 - (1+10/(10-(9-1)))(10^(10-(9-1))+1)

190 = 19*91 - 901 - (10+19)*(10+9^0)*(10-(9-1))

190 = (1+9+0)*19

190 = (1+90)*(10-(9-1)) + (9-1^0)

190 = 19*10

190 = (1*9+0)*19 + 19*1^0

190^2 has one 1 and no 9. 190^3 has one 9 and no 1.

190^4 has two 1's and no 9's. 190^5 has 2 9's and no 1's.

(190+1*9) and (190-1*9) are both prime.

190 = 1+2+3+4+5+...+19.

190^128 + 1 is prime.

190^16 + 191^16 is prime.

The 190th and 191st primes are twin primes.

190 = 4^4 - 3^4 + 2^4 - 1^4.

190^3 +/- 3 are consecutive primes.

There are 190 squares less than 33^3.

190 = 19 * 5 * 2

CXC = XIX * V * II

(Note all palindromes in Roman numerals).

July 10th (191st Day of the Non-Leap Year)

Jul 10th, 7/10.

710 = 701 + (10-7^0)

710 = 107 + 10*7 + (17-(10-7+1^0))*(71-(7-1)*(10/(7^0+1)))

710 = 170 + 1*70 + (71-(17+1*7))*10

710 = 71*10

710 = 71*17 - 701 + 17*(7-1)*(10-(7+1))

710 = (7-1)! - 10

710 = (10-(1+0))^(10-7)-(17+(10-(1+7)))

710 = (17+1*7)^(10-(7+1)) + (70-(10-7))*(10-(7+1))

710 = (7+10) + (71+0) + (0+17) + (7-1-1^0)*(10+1+0)^(10-(7+1))

710 = (17+(1+7))^(10-(7+1)) + 17*(10/(7^0+1))

710 = (7+1+0)*(71+17) + (7-1-0)

710 = (7+10)*(7*(7-1)) - (10-(7-1))

710^9 + 710^7 + 710^5 + 710^3 + 710^1 + 1 is prime.

710^710 - 710 - 1 is prime.

709^2 + 710^2 and 710^2 + 711^2 are both prime.

191st Day of the Year.

191 = 119 + 9*(9-1)

191 = 911 - (9+1)*9*(9-1)

191 = (19-(11-9))^(11-9) - (1+1)*(9-1-1)^(11-9)

191 = 19*1 + 1*91 + 9^(1+1)

191 = (1+91) + (19+1) + (91-(11+1*1))

191 = (1+9+1)*(19-(1+1)) - (11-(9-1-1))

191 = (19+1)*(1*9*1) + (1+9+1)

191 = (1+91)*(11-9) + (9-1-1)

191 = (1+9*1)*19 + (11-(1*1+9))

191, 193, 197, and 199 are all prime.

191^3+191+1 is prime.

191^3-191-1 is prime.

There are 191 primes less than 34^2.

There are 191 0's in the first 1000 primes.

191 cents is obtained from a 1 dollar coin, half-dollar coin, quarter, dime, nickel and penny.

A pancake cut 19 times can produce a maximum of 191 bites.

July 11th (192nd Day of the Non-Leap Year)

Jul 11th, 7/11.

711 = 171 + 1*71 + 17*1 + (11-7)*(117-(11-7))

711 = 117 + 1*17 + 11*7 + (7-1-1)*(11-1*1)^(11-(7+1+1))

711 = (7+1+1)^(11-(7+1)) - (17+1)

711 = (7-1*1)! - (7+1+1)

711 = 71*17 - 117 - 171 - (11-7)*(71-(17+1+1))

711 = (7+11) + (71+1) + (11+7) + (1+17) + (7-1-1)*117

711 = (7+1+1)*(71+(7+1))

711 = (71+1)*(11-1*1) - (7+1+1)

711 = (71-1)*(11-(1+1)) + (7+1+1)^(1+1)

711 = (7+11)*((7+1)*(7-1-1)) - (7+1+1)

711 = (7+1*1)*(71+17+1) - ((7-1-1)-(11-7))

711^2 contains 2 consecutive 5s.

711^4 contains 4 consecutive 5s.

711^5 is the first power of 711 > 1 that contains a 7.

711^11 + 711^9 + 711^7 + 711^5 + 711^3 + 711 +1 is prime.

711^15 + 711^13 + 711^11 + 711^9 + 711^7 + 711^5 + 711^3 + 711 + 1 is prime.

711 is a popular convenience store in the United States and Japan.

192nd Day of the Year.

192 = 129 + 9*(9-2)

192 = 219 - (12-9)^(9/(2+1))

192 = 291 - 9*(2+9*1)

192 = 912 - 9*12 - 91*2 - (9+1)*(29+(9-2)*2)

192 = 921 - 9*21 - 92*1 - (1+9)*(92-(29-1))

192 = (19-(9-2-1))^(12-(1+9)) + (29-(9-2-1))

192 = (2*(9-2))^(9/(2+1)-1) + (12-(9-1))

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

192 = 2*91 + 29*1 - (21-2*1)

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

192 = (2+91) + (29+1) + (9/(2+1))*(29-(9-2-1))

192 = (1+9+2)*(2^((9-1)/2))

192 = (1+92)*(21-19) + (9-2-1)

192 = (19+2)*9 + (12-9)

192 = (19-2)*(1*9+2) + (1+9)/2

192 = (1+9*2)*(1+9) + (21-19)

192^2 and 192^3 both do not contain a 1, 9, or 2.

192*(1*9*2) = 3456.

192^2+192+1 is prime.

There are no primes between 1920 and 1929.

192^6+192^5+192^4+192^3+192^2+192+1 is prime.

The sum of the first 192 primes is prime.

2^192 + 192^2 + 1 is prime.

192^16+193^16 is prime.

There are 192 5-digit left-truncatable primes.

192 is the smallest number with 14 divisors.

192, 192*2 (384) and 192*3 (576) all contain the digits 1 through 9 exactly once.

192 = 4*6*8 (product of first three composites).

July 12th (193rd Day of the Non-Leap Year)

Jul 12th, 7/12.

712 = 721 - (7+2*1)

712 = 127 + 12*7 + ((12-7)*(7+1+2)^2+1)

712 = 172 + 1*72 + 12*(27+12)

712 = 217 + 21*7 + (17+12)*12

712 = 271 - 2*71 + (17+(7-1))*(12+(2-1))

712 = 7*12 + 71*2 + 2*(17-2*7)^(12-7)

712 = 2*17 + 21*7 + (7*1+2)*(71-12)

712 = (7+12) + (71+2) + (7+1+2)*(72-(7+1+2))

712 = (2+17) + (21+7) + (7+12)*(7*(12-7))

712 = (7+1+2)*71 + (12/(7-1))

712 = (7*1+2)*(71+(7+1)) + (17-2^(7-1-2))

712 = (7+12)*(21+17) - (7+1+2)

712 = (71+2)*(7+1+2) - (7-1)*(1+2)

712 = (71-2)*(7+1+2) + (27-(7-2))

712 = (7*1*2)*(72-21) - (7*1*2-12)

712^2 does not contain a 7, 1, or 2.

The sum of the first 21 primes is 712.

(2*712)! / 712! +- 1 are twin primes.

+- 1^3 +- 2^3 +- 3^3 +- ... +- 23^3 +- 24^3 = 0 has 712 solutions.

(7!+k)/(7+k) is an integer when k = 712 (first k-value to work).

712 is the largest known number such that 712 and 712^8 (66,045,000,696,445,844,586,496) have no common digits.

193rd Day of the Year.

193 = 139 + 9*(19-13)

193 = 319 - 3*19 - (93-3*(9-1))

193 = 391 - 3*91 + 3*(9-3-1)^(9/3-1)

193 = 931 - 9*31 - 9*(39+(3+9))

193 = 913 - 91*3 - (139+(13-1*3))*3

193 = 1*93 + 19*3 + (39+(9/3+1))

193 = 3*91 - 39*1 - (39+(9/3-1))

193 = (1+93) + (19+3) + (9-(3-1))*(13-(3-1))

193 = (3+91) + (39+1) + (91-(9/3-1)^(9-3-1))

193 = (1+9+3)*(13+(3-1)) - (9/3-1)

193 = (19+3)*9 - (13-(9-1))

193 = (19-3)*(1*9+3) + (19-3*(9-3))

193 = (1+93)*(9/3-1) + (9-3-1)

193 = (1+9*3)*(9-(3-1)) - (1*9/3)

193 = (1*9-3)*31 + (9-(3-1))

193^5 ends in 193 (also smallest prime where p^5 contains all digits 1 - 9).

193^4+193^3+193^2+193+1 is prime.

193^4-193-1 is prime.

193/71 is the most accurate approximation with two primes < 2000 of e = 2.718281828...

July 13th (194th Day of the Non-Leap Year)

Jul 13th, 7/13.

713 = 731 - (7-1)*3

713 = 137 + 13*7 + (73+(7+1)*3)*(7-(3-1))

713 = 173 + 17*3 + (137+13*(7+1)/(3+1))*3

713 = 371 + 3*71 + (31+(7-1)*(3-1))*3

713 = 317 + 31*7 + (137+(7-1)*7)

713 = 7*13 + 71*3 + (371+(37+1))

713 = 3*17 + 31*7 + (7-(3-1))*(73+(13+1*3))

713 = (7+13) + (71+3) + 371 + 31*(7+1)

713 = (3+17) + (31+7) + 37*(13-(7-1))

713 = (7+1+3)*(13*(7-(3-1))) - (7+1)/(3+1)

713 = (7*1+3)*71 + (17-7*(3-1))

713 = (7+13)*(37-1) - (17-(7*1+3))

713 = (71+3)*(13-(1+3)) + (31+(3-1)^(7-3))

713 = (71-3)*(7*1+3) + (31+(3-1))

713 = (7*1*3)*(31+3*1) - (13-(7-1)*(3-1))

194th Day of the Year.

194 = 149 + (1+4)*9

194 = 419 - 41*9 + (4*(9/(4-1)))^(9/(4-1)-1)

194 = 491 - 4*91 + (94-(4-1)*9)

194 = 914 - 91*4 - 4*(94-(9-4))

194 = 941 - 9*41 - (14+1*4)*(41-4*(9-4))

194 = 1*94 + 19*4 + 4*(9-(4-1))

194 = 4*91 - 49*1 - (19-(9-1))^(9/(4-1)-1)

194 = (1+94) + (19+4) + 19*4

194 = (4+91) + (49+1) + 1*49

194 = (1+9+4)*14 - (9/(4-1)-1)

194 = (1*9+4)*(19-4) - (19-(4-1)*(9-(4-1)))

194 = (1+9*4)*(1*9-4) + (14-1*4)

194 = (19+4)*(9-1) + (14-1*4)

194 = (1+94)*(9/(4-1)-1) + (14-(9+1))

194^2 does not contain a 1, 9, or 4.

194 = 7^2+8^2+9^2.

194^2+194+1 is prime.

194^4+1 is prime.

194^16 + 195^16 is prime.

4567890123456789012345678901234567... (194 digits) is prime.

194 is 1234 in base 5.

194 is 365 in base 7 (365 days a year, 7 days a week ...).

July 14th (195th Day of the Non-Leap Year)

Jul 14th, 7/14.

714 = 741 - (4-1)^(7-4)

714 = (7-1)! - (17-(1+7))!

714 = 417 + 41*7 + (7-4-1)*(1+4)

714 = 471 + 4*71 - ((7+1)*(4+1)+1)

714 = 174 + 17*4 + (7+1)*(71-(7+1+4))

714 = 147 + 14*7 + (71-4)*7

714 = 71*4 + 7*14 + 4*(71+(7+1+4))

714 = 4*17 + 41*7 + ((14+(1+4))^(14/7)-(14/7))

714 = (7+14) + (71+4) + (7-1)*(147-4*(7+1*4))

714 = (4+17) + (41+7) + (47-4)*(4-1)*(4+1)

714 = (7+1+4)*(71-7*1-4) - (14-(7+1))

714 = (7*1+4)*(71-(7-1)) - (14/7-1)

714 = (71+4)*(14-(1+4)) + (41-(7-4-1))

714 = (7+14)*(17*(14/7))

714 = (71-4)*(7+1*4) - (17+(7-1))

1+1/2+1/3+1/4+..+1/714 has a prime numerator.

714 = 10324 in base 5 (one digit used each).

195th Day of the Year.

195 = (19-5)^((9-1)-(5+1))-1

195 = 159 - (1-5)*9

195 = 591 - 5*91 + 59*1

195 = 519 - 51*9 + (1*9*5)*(9-5-1)

195 = 951 - 9*51 - (9-5-1)*(91+(9-1))

195 = 915 - 91*5 - (59-(5+1))*5

195 = 1*95 + 19*5 - (1+9-5)

195 = 5*91 - 59*1 - (9-5-1)*(91-(19+5))

195 = (1+95) + (19+5) + 5*(19-(5-1))

195 = (5+91) + (59+1) + (19+5*(9-5))

195 = (1+9+5)*(15-(1+9)/5))

195 = (1*9+5)*(19-5) - ((1+9)/5-1)

195 = (1+9*5)*(1*9-5) + (19-(9-1))

195 = (19+5)*(9-1) + (9-5-1)

19*5 = 1*95 (any other nontrivial numbers like this?).

195^2+196^2 is prime.

195^4+196^4 is prime.

195^10+195^9+195^8+195^7+195^6+195^5+195^4+195^3+195^2+195^1+1 is prime.

195^39 + 195^37 + 195^35 + 195^33 + ... + 195^5 + 195^3 + 195^1 + 1 is prime.

July 15th (196th Day of the Non-Leap Year)

Jul 15th, 7/15.

715 = 751 - (7*5+1)

715 = 157 + 15*7 + 3*(175-(7+1)*(7-(5-1)))

715 = 175 + 17*5 + (1*7*5)*(7+1+5)

715 = 571 + 5*71 - (175+(1+7*5))

715 = 517 + 51*7 - 3*(57-(5-1))

715 = 7*15 + 71*5 + 51*(5*1)

715 = 5*17 + 51*7 + (51-(7+1*5))*7

715 = (7+15) + (71+5) + (571+(51-5*1))

715 = (5+17) + (51+7) + 5*(157-(7-1)*5)

715 = (7+1+5)*(57-(7-5))

715 = (7+1*5)*(57+(7-5)) + (15-(7+1))

715 = (7+15)*((7-5)^5) + (15-(5-1))

715 = (7*1*5)*(15+1*5) + 5*(7-(5-1))

715 = (71+5)*(15-(1+5)) + (17+7*(7-5))

715 = (71-5)*(7+(5-1)) - (17-(7-1))

715716 is a square.

715^23 + 715^21 + 715^19 + 715^17 + ... + 715^5 + 715^3 + 715^1 + 1 is prime.

5^5^5^5^5^5^5^5^5^5 has 715 distinct values depending on parentheses placement.

715^32 + 716^32 is prime.

196th Day of the Year.

196 = 169 + 1*9*6/(9-6-1)

196 = (16-(9-6-1))^(9-6-1)

196 = 619 - 61*9 + (1+6)*(16+(9-6-1))

196 = 691 - 6*91 + (69-(19-1))

196 = 961 - 9*61 - 6^(9-6)

196 = 916 - 91*6 - (19+(1+9))*6

196 = 1*96 + 19*6 - (19-(6-1))

196 = 6*91 - 69*1 - (169+16*(1+6))

196 = (1+96) + (19+6) + (61+(19-6))

196 = (6+91) + (69+1) + (19+(1+9))

196 = (1+9+6)*(19-6) - 6*(9-6-1)

196 = (1+9*6)*(9-6*1) + (16+(6-1)*(9-6))

196 = (1*9+6)*(19-6) + (16-(1*9+6))

196 = (1+96)*(9-6-1) + (9-6-1)

196 = (19+6)*(9-1) - (9-(6-1))

The sum of the first 196 primes divides the product of the first 196 primes.

196 + 691 = 887.

887 + 788 = 1675.

1675 + 5761 = ...

196 is the smallest conjectured number such that the right hand side of the equation will never be a palindrome.

Numbers like this are Lychrel numbers.

July 16th (197th Day of the Non-Leap Year)

Jul 16th, 7/16.

716 = 761 - (61-16)

716 = 671 + 67*1 - (16+1*6)

716 = 617 + 6*17 + (16-(7*1+6))

716 = 167 + 16*7 + (7+16)*(7+6*(7-(6-1)))

716 = 176 + 17*6 + (71+(7-(6-1)))*6

716 = 7*16 + 71*6 + (7-(6-1))*(76+(7+6))

716 = 6*17 + 61*7 + 17*(17-6)

716 = (7+16) + (71+6) + (76+1)*(16/(7-(6-1)))

716 = (6+17) + (61+7) + (6-1)^(17-(7*1+6))

716 = (7+1+6)*(17*(17-7*(7-(6-1)))) + (7-(6-1))

716 = (7*1+6)*(61-6*1) + (17-16)

716 = (71+6)*(16-7) + (7+16)

716 = (71-6)*(17-6) + (7*1-6)

716 = (7+16)*(67-7*(6-1)) - (16+(7+1)/(7-(6-1)))

197th Day of the Year.

197 = 179 + (19-(9-7-1))

197 = 719 - 7*19 - 17^(9-7) - (1+9)^(19-17)

197 = 791 - 79*1 - (19-7*(9-7))*(97+(7-1))

197 = 971 - 9*71 - (17-(7-1)*(9-7))*(19+(9-1))

197 = 917 - 91*7 - (91-(9-1))

197 = 1*97 + 19*7 - (9-(7-1))*(17-(7-1))

197 = 7*91 - 79*1 - 19^(9-7)

197 = (1+97) + (19+7) + (79-(7-1))

197 = (7+91) + (79+1) + (97-79+1)

197 = (1+9+7)*(19-7) - (17-(1+9))

197 = (1*9+7)*(19-7) + (9+1)/(9-7)

197 = (1+9*7)*(9-(7-1)) + (19-(9-7)*7)

197 = (19+7)*7 + (17-(9-7))

197 = (1+97)*(9-7) + (9-7-1)

197^2-197-1 is prime.

197 is the smallest prime that is the sum of 7 consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41).

197 is the sum of the first 12 primes (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37).

The two-digit primes are {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.

Adding up all of their digits, you get 197.

111+9+77 = 197.

197 = 4^2 + 6^2 + 8^2 + 9^2 (squares of first four composites).

July 17th (198th Day of the Non-Leap Year)

Jul 17th, 7/17.

717 = 771 - (77-(17+(7-1)))

717 = (7-1)! - (17-7*(7/7+1))

717 = (17-7*(1+7/7))^(7-1) - (7-1)*(7/7+1)

717 = 177 + 17*7 + (17+7)*17 + (7+7-1)

717 = 177 + 1*77 + 71*7 - 17*(1+7/7)

717 = 7*17 + 71*7 + 7^(7/7+1) + (7/7+1)*(17+(7+7/7+1))

717 = 77*1 + 7*1*7 + 177 + (17+1)*(17+(7-1))

717 = (7+17) + (71+7) + (7*1*7-(1+7))*(7+1+7)

717 = (77+1) + (7+1+7) + (17+7)*(17+(7+7/7+1))

717 = (7+1+7)*(7*7-1) - (17-(7+7*1))

717 = (7*1+7)*(7*7+1) + 17*(7/7*1)

717 = (71+7)*(17-(1+7)) + (7+1+7)

717 = (71-7)*(17-(7-1)) + (7+7-1)

717 = (7+17)*(7+1+7)*(1+7/7) - (7+1+7)/(1+7/7+1)

717 = 3*239 and 3+239 = 242, also a palindrome.

3^717 is the smallest exponent of 3 containing exactly 48 of the same digit.

717^3 +/- 4 is prime.

717*717! - 1 is prime.

A popular US airplane in the Boeing 717.

198th Day of the Year.

198 = 189 + (18-9)

198 = 918 - 91*8 + (89-81)

198 = 981 - 9*81 - (9-(8-1))*(19+8)

198 = 819 - 8*19 - (8-1)*(81-(9-(8-1))*(8-1))

198 = 891 - 89*1 - (9-(8-1))*(189+(8/(9-(8-1)))*(91-(19*(9-(8-1)))))

198 = (1+98) + (19+8) + 1*9*8

198 = (8+91) + (89+1) + (18-9)

198 = (1+9+8)*(19-8)

198 = (1*9+8)*(18-(8-1)) + (19-8)

198 = (1+9*8)*(1+18/9) - (19+(9-(8-1)))

198 = (19+8)*(8-1) + (18-9)

198 = (1+98)*(9-(8-1))

198^5 is the next power of 198 > 1 that contains an 8.

198^4+1 is prime.

198^16+1 is prime.

198^4+198^3+198^2+198^1+1 is prime.

The sum of the first 198 primes is prime.

198^4-198-1 is prime.

198^8 + 199^8 is prime.

There are 198 nonzero palindromes < 10000.

999...9991 (198 9's) is prime.

198 = 3012 in base 4 (uses all possible digits).

198 = 19//8 (concatenation function). Further, 19 is the 8th prime number and 198 is the (19*8)nd composite.

July 18th (199th Day of the Non-Leap Year)

Jul 18th, 7/18.

718 = 781 - 7*(8+1)

718 = 871 - 87*1 - (18-7)*(7-1)

718 = 817 - 8*17 + (71-17*(8-(7-1)))

718 = 178 + 17*8 + 1*78 + (8-(7-1))*(187-(17+7))

718 = 187 + 18*7 + 1*87 + (7-1)*(8*7-(18/(7-1)))

718 = 7*18 + 71*8 + (17+1*7)

718 = 8*17 + 81*7 + (7*1+8)

718 = (7+18) + (71+8) + 187 + 7*(8*7+(7-1))

718 = (8+17) + (81+7) + (8-18/(7-1))*(18-7)^(8-(7-1))

718 = (7+1+8)*((7*1+8)*(18/(7-1))) - (8-(7-1))

718 = (7*1+8)*(7*(8-1)) - (18-(8-7))

718 = (71+8)*(18/(8-(7-1))) + (78-71)

718 = (71-8)*(18-7) + (7+18)

718 = (7+18)*(18+(18-7)) - (78-71)

718^5 is the next power of 718 > 1 that has an 8.

There are 718 distinct values for 6^6^6^6^6^6^6^6^6^6 (depending on parentheses).

718^25 + 718^23 + 718^21 + ... + 718^5 + 718^3 + 718^1 + 1 is prime.

199th Day of the Year.

199 = 919 - 9*19 - (9*9-(19+9/9))*9

199 = 991 - 99*1 - 9*(91-(1+9/9)*(9-(1+9/9)))

199 = 1*99 + 19*9 - (99-(19+9))

199 = 9*91 - 99*1 - 1*9*9 - (1+9)*(19+(19-(9-(1+9/9))*(1+9/9))^(9/9+1))

199 = (1+99) + (19+9) + (99-(19+9))

199 = (9+91) + (99+1) - 1*(9/9)

199 = (1+9+9)*(1+9) + (99/(9+9/9+1))

199 = (1*9+9)*(19-(9-1)) + 9/(1*9)

199 = (1+9*9)*(1+9/9) + ((1+9)/(1+9/9))*(9-(1+9/9))

199 = (19+9)*(9-(1+9/9)) + ((1+9)/(1+9/9)-(1+9/9))

199 = (1+99)*(1+9/9) - (9-(9-1))

199^3, 199^5, 199^7, ... all end in 99.

199, 919, 991 are all prime.

199^2+200^2 is prime.

199^16+200^16 is prime.

The next prime is 211. 199211 and 211199 are both prime.

"Just average 101 and 197 to get this prime number" was a clue in the TV show Jeopardy!, February 1st, 2010.

199 is the first in a chain of near-repdigit consecutive primes: 199, 211, 223, 227, 229, 233.

1000^199 - 999^199 is prime.

July 19th (200th Day of the Non-Leap Year)

Jul 19th, 7/19.

719 = 791 - (7+1)*9

719 = 197 + 1*97 + 17*(17+(1+7))

719 = 179 + 1*79 + 17*9 + (71+(7-1))*(19-(9+7-1))

719 = 971 - 9*71 + (97-9*(7-1))*9

719 = 917 - 91*7 + 197 + (9-7)*(17-(7-1))^(9-7)

719 = 7*19 + 71*9 - (9*(7-1)-1)

719 = 9*17 + 91*7 - (9*7+(17-9))

719 = (7+19) + (71+9) + 179 + (7*(9-7))*(19+(7-1)*(9-7))

719 = (9+17) + (91+7) + 17*(19+(7*1+9))

719 = (7+1+9)*(7*(7-1)) + (19-7*(9-7))

719 = (7*1+9)*(71-(7+19)) - (9-7-1)

719 = (7+19)*(19+(9-1)) + 17*(9-7-1)

719 = (71+9)*9 - (9-7-1)

719 = (71-9)*(19-7) - (17+(1+7))

719^3-719-1 is prime.

There are 719 distinct values for 7^7^7^7^7^7^7^7^7^7 (depending on parentheses).

Also, if you have 8, 9, or 10, instead of 7, there are also 719 distinct values.

2357131719...719 is prime (Concatenating the primes).

e^(pi*sqrt(719)) is less than .0001 from an integer.

The hour and second hand align 719 times every 12 hours.

n^4 + 179 is composite for n = 1, 2, 3, ... 719.

200th Day of the Year.

200 = 20*2*(20/(2+2^0+2^0))

200 = 20^2/2

200 = (20-2*(2+2^0))^2 + 2*(2+0+0)

200 = ((2+2^0)*(2+2+2/2))^2 - (20/(2+2))^2

There are 200 primes < 35^2.

The sum of the first 200 primes divides the product of the first 200 primes.

July 20th (201st Day of the Non-Leap Year)

Jul 20th, 7/20.

720 = 702 + (7-2^0)*(2+7^0)

720 = (7-2^0)!

720 = 72*(7+2+2^0)

720 = 270 + 2*70 + (20/2)*(27+20/(7-2))

720 = 207 + 20*7 + 2*70 + (270-(70/2+2))

720 = 7*20 + 72*(2+0) + (7-2-2^0)*(72+27+(7+2+2^0))

720 = 27*20 + (70+20)*2

720 = 72*27 - 702 - (27-(2+7))*(27+2)

720 = (7+2+0)*(20*(7-2-2^0))

720 = (7*2+0)*(27*2) - (7-2^0)^2

720 = (7+20)^2 - (7+2+0)

720 = (3!)! (next one is 24 digits long).

720 = 6! = 5!*3!

720 = 1*2*3*4*5*6 = 8*9*10.

A credit score of 720 or higher is deemed a good credit score.

201st Day of the Year.

201 = 210 - (10-2^0)

201 = 120 + (10*(20-1)-10^2)

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

201 = 20*1 + (2+0+1) + 2*(102-(12+2^0))

201 = 21*12 - (2+0+1)*(20-(2+0+1))

201 = (20+1) + (21+0) + (2+0+1)*(21+2^(10/2))

201 = (10+2) + (12+0) + (10+(10-(2+0+1))^2)*(2+0+1)

201 = (2+0+1)*(12+(10/2)*(12-(2-1)))

201 = (20+1)*10 - (10-(1+0))

201 = (20-1)*10 + (12-(2-1))

201^11 + 201^9 + 201^7 + 201^5 + 201^3 + 201^1 + 1 is prime.

1+1/2+1/3+...+1/201 has a prime on the numerator.

The 201st and 202nd prime are twin primes.

July 21st (202nd Day of the Non-Leap Year)

Jul 21, 7/21.

721 = 712 + (7*1+2)

721 = 271 + 27*1 + (7+2*1)*(21+(27-1))

721 = 217 + 2*17 + (7+2+1)*(7^2-2*1)

721 = 127 + 12*7 + 17*(21/7)*(7+2+1)

721 = 172 + 1*72 + (27*2-1)*(7*1+2)

721 = 7*21 + 72*1 + 217 + (17+2)*(7*2+1)

721 = 1*27 + 12*7 + (7+2+1)*(71-(7+2+1))

721 = (7+21) + (72+1) + (71-(7+2))*(7+2+1)

721 = (1+27) + (12+7) + 271 + (7*2-1)*(27+(7-2-1))

721 = (7+2+1)*72 + (21/7-2*1)

721 = (7*2+1)*(7^2-1) + (21-7)/(7*2*1)

721 = (7+2*1)*((7+2)^2-1) + ((7*2-1)-12)

721 = (7*2*1)*(17*(2+1)) + 7*(2-1)

721 = (7+21)*(27-1) + 7*(1-2)

721 = (72-1)*(7+2+1) + (17-(7-1))

721 = (72+1)*(7+2+1) - (7+2*1)

There are 721 pairs (a,b,c) for -15 <= a, b, c <= 15 where a+b+c = 0.

721^2 = 519841 = 228^2//1^2 (// is the concatenation function).

721^3 - 721^2 +/- 1 are twin primes.

3^721, 3^722, 3^723 all have the same number of digits.

If 721 = n, 101001000n is prime (Meaning, 101001000721 is prime).

202nd Day of the Year.

202 = 220 - (20-2)

202 = 22^2/2 - 20*2

202 = 20^2/2 + 2^(2^0)

202 = (2+0+2)*20*2 + (2*(2+2^0))*(20/2-(2+2^0))

202 = 22*(20-2)/2 + (2+0+2)

202 = 20*(20-2)/2 + (20+2)

202 = 20*22/2 - (20-2)

202 = (20-2)^2/2 + 2*20

202 = (20/2)*20 + (2^0)*2

202^4 does not contain a 0 or 2.

202^2 + 203^2 is prime.

10^202+9 = 100...009 (201 0's) is prime.

July 22nd (203rd Day of the Non-Leap Year)

Jul 22nd, 7/22.

722 = 227 + 2*27 + (7*(7-2-2))^2

722 = 272 + 2*72 + (7+2)*(27+7)

722 = 7*22 + 72*2 + (27*2-2/2)*(7+2/2)

722 = 2*27 + 22*7 + 227 + 7*(27+2*7)

722 = (7+22) + (72+2) + 272*2 + (7-2)*(22-7)

722 = (2+27) + (22+7) + 2^(7-2-2)*(72+(7+2+2))

722 = (7+2+2)*(72-(7-2)) - (22-7)

722 = (7*2+2)*(7^2-2) - (7-2)*(7-2/2)

722 = (7+22)*(27-(7-2-2)) + (27-2/2)

722 = (72+2)*(7+2) + 7*(22-7*2)

722 = (72-2)*(7+2+2) - (7^2-2/2)

722*227 does not contain a 2 or 7.

+- 1 +- 2 +- 3 +- ... +- 15 = 0 has 722 solutions.

2^722 contains exactly 36 of the same digit.

203rd Day of the Year.

203 = 230 - 3^(2+3^0)

203 = 302 - 3*(32+2^0)

203 = 320 - 3*(20*3-(23-2))

203 = 20*3 + 30*2 + ((2^3)*(20/2)+3)

203 = 20*30/2 - 3^2*(20-3^2) + (3-2^0)

203 = 20^(3-2^0)/2 + (30/(2*(2+0+3)))

203 = 23*(2+3) + (2^3)*(20-3^2)

203 = 32*(3+2) + (20+23)

203 = 23*(2*3) + (2+3)*(2+(20-3^2))

203 = 32*(3*2) + (20-3^2)

203 = (2+0+3)*(20*2) + (3+2*0)

203^3 does not contain a 0.

203^3 contains {2,3,4,5,6,7,8} exactly once.

203^4 does not contain a 2, 0, or 3.

203^2+203 +/- 1 are twin primes.

203^6 + 203^5 + 203^4 + 203^3 + 203^2 + 203^1 + 1 is prime.

203^3-203-1 is prime.

July 23rd (204th Day of the Non-Leap Year)

Jul 23rd, 7/23.

723 = 732 - (2^(7-3)-7)

723 = 372 + 37*2 + 3*72 + (73-(7+3+2))

723 = 327 + 32*7 + (27-23)*(7^2-2*3)

723 = 273 + 2*73 + (3^2+7)*(27-2^3)

723 = 237 + 23*7 + (7+2*3)*(7*3+(7-3))

723 = 7*23 + 72*3 + (372-(27-(7-3*2)))

723 = 3*27 + 32*7 + (27-(7-2))*(7*3-2)

723 = (7+23) + (72+3) + (7-(3-2))*(2^7-(7-2)^2)

723 = (3+27) + (32+7) + (33-27)*(2^7-(7*3-2))

723 = (7+2+3)*(72-(7+3+2)) + ((7-3)*(3+2)+3)

723 = (7*2+3)*(7*2*3) + 27/3

723 = (7+2*3)*(7*2^3) - (37-32)

723 = (7+23)*(27-3) + (7+2)/3

723 = (72+3)*(7*2-(2+3)) + (27-3)*(3^2-7)

723 = (72-3)*(23-(7+2*3)) + 3*(7*2-3)

723 = 3*241 and 723*3241 = 2343243.

There are 723 primes between 54^3 and 55^3.

723^2 +/- 2 are both semiprimes (as is 723).

723^4//723^3//723^2//723^1//723^0 is prime (Meaning, 2,732,456,074,41|3,779,330,67|5,227,29|7,23|1 is prime).

723^30 + 723^29 + 723^28 + ... + 723^3 + 723^2 + 723^1 + 1 is prime.

204th Day of the Year.

204 = 240 - 4*(4*2+42^0)

204 = 420 - 4*20 - 4*(40-(2+0+4))

204 = 402 - 40*2 - (2^4*(2+0+4)-20/(4-2))

204 = 20*4 + (20/4)*(2^4+4^2) - (4+2)^(4-2)

204 = (20+4) + (40+2) + (2+0+4)*(24-24^0)

204 = 24*(2+4) + 20*(4-2^0)

204 = 42*(4+2) - (42+(4+2))

204 = 24*42 - 420 - 402

204 = 24*(2*4) + 2*(2+0+4)

204 = 42*(4*2) - (2+0+4)*(2^4-20/4)

204 = (2+0+4)*(2+2^(4+2^0))

204 = (20-4)*(20-2*4) + 2*(2+0+4)

204*204 = 41616 (concatenation of 3 squares).

204^5 is the next highest power of 204 > 1 that contains a 0 or a 2.

204 = 2*2*3*17 and 204*(2+2+3+17) = 4896 (first four composite numbers).

204 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2.

204^2 + 1 is prime.

204^4 + 1 is prime.

204^4 + 204^3 + 204^2 + 204^1 + 1 is prime.

The sum of the first 204 primes is prime (also for 206, and 208 -- a run of three).

There are 204 ways to place 3 nonattacking queens on a 5 by 5 board.

204^4 + 204 + 1 is prime.

204^5 - 204 - 1 is prime.

If 204 = n, then 2n3n5n7n11n13 is prime (Meaning 22043204520472041120413 is prime).

1000..000999...999 (with 204 0's and 204 9's) is prime.

204 = 101 + 103 and 204^2 = 41616 = 20807 + 20809 (sum of twin primes).

If a knight moves 8 times, there are 204 different squares it could end up.

In poker, there are 204 hands that are at least as good as a straight flush (allowing for 1 joker).

July 24th (205th Day of the Non-Leap Year)

Jul 24th, 7/24.

724 = 742 - (7*2+4)

724 = 274 + 27*4 + (7*2+4)*(27-2*4)

724 = 247 + 24*7 + (7-4)*(7*2*4+47)

724 = 427 + 42*7 + 24/(2*4)

724 = 472 + 47*2 + (72+7)*2

724 = 7*24 + 72*4 + (72-(7-2))*4

724 = 4*27 + 42*7 + (7*2)*(27-4)

724 = (7+24) + (72+4) + (7+2+4) + 4*(4*7^2-(47-2))

724 = (4+27) + (42+7) + (27-4)*(7*2*4/2)

724 = (7+2+4)*(7*2*4) - (7*4-24)

724 = (7*2+4)*(7^2-(7+2)) + (27-(7+2^4))

724 = (7+2*4)*(42+(4+2)) + (7*4-24)

724 = (7+24)*(27-4) + (24-(7+2+4))

724 = (72+4)*(7*2-(7-2)) + 4*(7*2-4)

724 = (72-4)*(7*2-4) + (47-(7-4))

724^n contains a 7, 2, and 4 for all n > 0.

There are 724 ways to place 10 nonattacking queens on a 10 by 10 board.

724^4//724^3//724^2//724^1//724^0 is prime.

205th Day of the Year.

205 = 250 - 5*(20/5+5)

205 = 502 - 50*2 - (20-5-2^0)^2 - (5-2^2)

205 = 520 - 5*20 - (52-(5*2+2^0))*5

205 = 20*5 + 2*(20+5) +5*(20/5+(2+0+5))

205 = 25*(2+5) + 5*(20/5+2)

205 = 52*(2+5) - (5-2)*(52+52^0)

205 = 52*(5-2) + (2+0+5)^2

205 = 52*25 - 502 - 520 - (52+(25-20/5))

205 = (2+0+5)*(25+20/5) + (20/5-2)

205 = (20+5)*(2^(5-2)) + (20/5+2^0)

205 = (20-5)*(20-5-2) + (2*5+0)

205^7 (17 digits) does not contain a 0.

There are 205 twin prime pairs < 10^4.

205 = 4^4 - 4^3 + 4^2 - 4^1 + 4^0.

2^1 + 0^2 + 5^3 is prime.

July 25th (206th Day of the Non-Leap Year)

Jul 25th, 7/25.

725 = 752 - (5-2)^(5*2-7)

725 = 527 + 52*7 - 2*(57+(27-(7/(2+5))))

725 = 572 + 57*2 + (5-2)*(27-2*7)

725 = 257 + 25*7 + ((7+2*5)^2+(25-7*(5-2)))

725 = 275 + 2*75 + (5-2)*(25*(7-(5-2)))

725 = 7*25 + 72*5 + (7*2+5)*(2*5)

725 = 5*27 + 52*7 + 257 - (25+(5-2)!)

725 = (72+5) + (7+25) + (72/(7+2))*(72+5)

725 = (5+27) + (52+7) + 2*(275+7*(5-2)!)

725 = (7+2+5)*52 - (75-72)

725 = 2*(7*2+5)^2 + (52-7^2)

725 = (7+2*5)*(7*(5-2)!) + (25-7*2)

725 = (72+5)*(7*2-5) + (7+25)

725 = (72-5)*(25-7*2) - (27-2*7-7/(2+5))

725 = (7+25)*(27-5) + 7*(5-2)

There are 725 squares < 2^19.

206th Day of the Year.

206 = 260 - 6*(6+6/2)

206 = 620 - 6*20 - 6*(62-(20-6-20^0))

206 = 602 - 60*2 - (2*6)*(26-6/2)

206 = 20*6 + 26*(2+0) + 2*(20-6/2)

206 = 20*(6+2) + 2*(26-6/2)

206 = 20*(6-2) + 60*2 - (26+(2+6))

206 = 60*(6+2) - (260+(20-6))

206 = 60*(6-2) - 2*(20-6/2)

206 = (2+0+6)*26 - (6/(2+6^0))

206 = 26*62 - 620 - 602 - (6+0+2)*(26-6/2)

206 = (20-6)^2 + (26-2^(6-2))

206^2 + 1 is prime.

206^2 + 206 +/- 1 are twin primes.

206^6 + 206^5 + 206^4 + 206^3 + 206^2 + 206^1 + 1 is prime.

206^3 - 206 - 1 is prime.

The 206th and 207th prime are twin primes.

206^4 + 206 + 1 is prime.

206^2 + 207^2 + 208^2 is prime.

206 is the smallest number that, when written out in English, contains each vowel exactly once.

There are 206 bones in a typical adult human body.

July 26th (207th Day of the Non-Leap Year)

Jul 26th, 7/26.

726 = 762 - (76-(7*6-2))

726 = 267 + 26*7 +(276-(7-(6+2)))

726 = 276 + 2*76 + ((6/2)*(72+27)+(2-(7-6)))

726 = 627 + 6*27 - 7*(6*2-6/2)

726 = 672 + 67*2 - (7-2)*(6*2+(6-2))

726 = 7*26 + 72*6 + (6*2+(6-2))*7

726 = 6*27 + 62*7 + (7+6)*(6^2-26)

726 = (7+26) + (72+6) + (7+2+6)*(27+2*7)

726 = (6+27) + (62+7) + 26*(27-6/2)

726 = (7+2+6)*7^2 - (72/(2+6))

726 = (7*2+6)*6^2 + (27-7*(6/2))

726 = (7+2*6)*(26+2*6) + (7-6/2)

726 = (72+6)*(6*2-6/2) + (27-(6/2))

726 = (72-6)*(7*2-6/2)

726 = (7+26)*(27-(7-2))

726^n contains a 7, 2, and 6 for all n > 0.

7*2*6 = 84. 726*84 ends in 84.

726^32 + 1 is prime.

If 726 = n, then 2n3n5n7n11n13 is prime (Meaning, 27263726572677261172613 is prime).

726^2 + 726 + 1 is prime.

207th Day of the Year.

207 = 270 - 7*(2+0+7)

207 = 720 - 7*20 - (7-2)*72 - (20-7)

207 = 702 - 70*2 - (7-2)*(72-72^0)

207 = 20*7 + 70*2 - (72+72^0)

207 = 27*(7-2) + (72+0)

207 = 72*(7-2) - (7+2)*(20-(2+7^0))

207 = 27*(7+2) - (7-2^0)^2

207 = 72*(7+2) - (7*(2+7^0))^2

207 = 72*27 - 720 - 702 - (7-2)*7*(7+2)

207 = (2+0+7)*(2*7+(2+7))

207 = 27*7 + (20-(2+0))

207 = 72*(2+7^0) - (2+0+7)

207^2 contains a 2 but not 0 or 7.

207^3 contains a 7 but not 2 or 0.

207^4 contains a 0 but not 2 or 7.

Start with 1, 2, 3, 4, 5, 6, ... Remove the 2nd entry.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ... Remove the 3rd entry.

1, 3, 7, 9, 13, 15, 19, 21, 25, 27, ... Remove 4th entry.

1, 3, 7, 13, 15, 19, 25, 27, 31, ... Remove 5th entry.

...and so on. 207 is one that remains.

207^3 + 207 + 1 is prime.

207^4 - 207 - 1 is prime.

There are 207 distinct values for the 3rd derivative of x^x^...^x (25 x's) at x = 1.

July 27th (208th Day of the Non-Leap Year)

Jul 27th, 7/27.

727 = 772 - (7-2)*(7+2)

727 = 277 + 2*77 + (7+2+7)*(7*7-(7+7-2))/2

727 = 7*27 + 72*7 + (7+27)

727 = 27*(7-2) + 72*(7+2) - 7*(72/(7+2))

727 = 72*(7-2) + 27*(7+2) + 2*(72-(7+2+7/7))

727 = (7+2+7)*(27+(27-(2+7))) + (72/(7+2)-7/7)

727 = 77*2 + 7*2*7 + (27-2)*(7*2+(7-2))

727 = (7+27) + (72+7) + 2*(277+(27+2+7/7))

727 = (72+7)*(7+2) + (7+2+7)

727 = (72-7)*(27-(7+2+7)) + (7+7-2)

727 = (7+27)*(27-7) + (7*7-2)

727^4 is the next power of 727 > 1 that contains a 7.

1/727 repeats after 726 digits (longest possible).

Let a_0 = A, a_1 = B, and a_n = a_(n-1) + a_(n-2)

(For A = B = 1, the sequence is the Fibonacci sequence).

For any A and B, 727 divides at least one a_n value.

727^2 = 528529 (concatenation of two consecutive numbers).

733 is the next prime after 727. 727733 is also prime.

727, 733, 739, 743, 751, 757, 761, and 769 is the smallest run of 8 consecutive primes

such that all reversed are still prime.

727 is the smallest prime of the form n^3 - 2.

727 = 1! + (1+2)! + (1+2+3)!

727, 72727, 727272727, and 72727272727272727 are all prime.

(Flip previous term and self-concatenate.)

208th Day of the Year.

208 = 280 - 2*(8-2)^2

208 = 820 - 8*20 - (8/2)*(80+(20-8-2^0)*(8-2)/2)

208 = 802 - 80*2 - (20-(8-2))*(2*8*2-8^0)

208 = 28*(8-2) + (28+(20-8))

208 = 82*(8-2) - (8/2)*(82-(20-8-2^0))

208 = 82*(8+2) - 280 - (8/2)*(82+2^0)

208 = 28*(8*2) - (2+0+8)*(28-8/2)

208 = 2*80 + 8*20 - (2*8)*(8-2^0)

208 = (20+8) + (80+2) + 2*(8-2^0)^2

208 = (2+0+8) + (28+82) + 8*(20-(8+2^0))

208 = (2+0+8)*(8-2^0)*(2+8^0) - (8/2-2)

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

208 = (20-8)*(20-8/2) + 2^(8/2)

208^6 is the next power of 208 > 1 that contains a 0.

2081, 2083, 2087, and 2089 are all primes.

208^6 + 208^5 + 208^4 + 208^3 + 208^2 + 208^1 + 1 is prime.

208^4 + 208^3 + 208^2 + 208^1 + 1 is prime.

The sum of the first 208 primes is prime.

208^8 + 1 is prime.

7^3 - 6^3 + 5^3 - 4^3 + 3^3 - 2^3 + 1^3 = 208.

There are 208 squares < 35^3.

3^208 - 208 is prime.

208 is the heaviest stable isotope of any element (Lead).

There are 208 stitches in an average baseball.

July 28th (209th Day of the Non-Leap Year)

Jul 28th, 7/28.

728 = 782 - (7+2)*(8/2)

728 = 287 + 28*7 + (7-2)*7^2

728 = 278 + 2*78 + (8-2)*7^2

728 = 872 - 87*2 + 2*(7+8)

728 = 827 - 8*27 + (72/8)*(7*2-(8-7))

728 = 7*28 + 72*8 - (28+2*8)

728 = 8*27 + 82*7 - (78-2*8)

728 = (7+28) + (72+8) + ((27+(2+7))*(7+2+8)+(2-(8-7)))

728 = (8+27) + (82+7) + (28/7)*(7*2*8+(78/2))

728 = (7+2+8)*(7*(8-2)) + (28-2*8)

728 = (7+2*8)*(27+8/2) + (2-(7-8))*(7-2)

728 = (7*2+8)*(28+8/2) + (28-8/2)

728 = (72+8)*(7+2) + (72/(7+2))

728 = (72-8)*(2*8-8/2) - 8*(7-2)

728 = (7+28)*(28-7) - 28/(8/2)

728 = 2*2*2*7*13 and 728 is divisible by 2+2+2+7+13.

728^6 is the smallest power of 728 > 1 that contains a 7.

728^64 + 1 is prime.

728 is the smallest number that can be written as the sum/difference of two cubes in 3 ways.

("Least number written as a sum/difference of two cubes in N ways" are known

only up to N = 10).

209th Day of the Year.

209 = 290 - 9^2

209 = 920 - 9*20 - 9*(90/2+2*(9-2))

209 = 29*(2+9) - (90+20)

209 = 92*(9-2) - 29*(29+29^0)/2

209 = 29*(9-2) + (9-(2+9^0))

209 = 92*(9+2) - (2+0+9)*(92-(20-2^0))

209 = 20*9 + 29

209 = 92*29 - 920 - 902 - (20-(9-2))*(9-2)^2

209 = (2+0+9)*(20-9^0)

209^2 does not contain a 2, 0, or 9.

209 = 19*11 and 1911*209 = 399399 (concatenation of the same number).

209^2 + 209 +/- 1 are twin primes.

209th and 210th primes are twin primes.

exp(209) has the largest fractional part than any other exponent < 209.

209 =  1^+ 2^+ 3^+ 4^+ 5^+ 6^1.

77...777209 is prime (209 7's).

The sum of the first 209 composites is prime.

209 is a semiprime. Also 209^2 +/- 2 are both semiprimes.

July 29th (210th Day of the Non-Leap Year)

Jul 29th, 7/29.

729 = 792 - (7+2)*(9-2)
729 = 297 + 29*7 + (7*2*9+(92+(9+2)))
729 = 279 + 2*79 + (9-(7-2))*(72+(9-7)/2)
729 = 927 - 9*27 + 9*(7-2)
729 = 972 - 97*2 - 7*(9-2)
729 = 7*29 + 72*9 - 2*(9*7-2)
729 = 9*27 + 92*7 - 2*79
729 = (7+2+9)*(72+9)/2
729 = (7+2*9)*29 + (9-(7-2))
729 = (7*2+9)*(27+(7-2)) - (79-72)
729 = (7+29)*(72+9)/(9-(7-2))
729 = (72-9)*(9*2-7) + 9*(9-(7-2))

729 = 3*3*3*3*3*3 and 7+2+9 = 3+3+3+3+3+3.

729^2 does not contain a 7, 2, or 9 but 729^3 contains a 7, 2, and 9 once.

2^729+1 is divisible by 729.

729 = 6! + 3! + 2! + 1!

9!+k is first a square when k = 729.

729 = 3^(3!).

210th Day of the Year.

210 = 201 + 10-(2-1-0)
210 = 102 + 12*(10-(1+0))
210 = 120 + (12-1*2)*(10-(1-0))
210 = (2+1+0)*(21*(2+1)+(10-(2+1+0)))
210 = 21*(2+2^(2+1+0))
210 = 12*(20-(2+1+0)) + 2*1*(2+1)

210 = 2*3*5*7 (first four primes) (Note: 2+3+5+7 is prime).

210 = 1 + 2 + 3 + ... + 20

210^2 + 1 is prime.
210^4 + 1 is prime.

There are 210 primes less than 36^2 or 6^4.

210^23 + 210^21 + 210^19 + 210^17 + 210^15 + 210^13 + 210^11 + 210^9 + 210^7 + 210^5 + 210^3 + 210 + 1 is prime.

210!!!!!! + 1 is prime.

210 = 13+17+19+23+29+31+37+41 (8 consecutive primes).

July 30th (211th Day of the Non-Leap Year)

Jul 30th, 7/30.

730 = 703 + (37-(3+7))
730 = 307 + 30*7 + 3*(70+3^0)
730 = 370 + 3*70 + (3+7+0)*(7*3-(7-3^0))
730 = 73*(7+3+0)
730 = 73*(7-3) + (7-3^0)*73
730 = 37*(7-3) + 3!*(73+3*(7+3^0))
730 = 73*37 - 730 - 703 - 370 - (7+3^0)*7*3
730 = 37*(3*7) - (37+(3+7+0))
730 = 73*(7*3) -703 - (7+3+0)^(3-7^0)

730 = 2*5*73 (uses all prime digits).

730 = (2^3+1)^3+1.

730^4 + 1 is prime.

730 cannot be expressed as a square plus a prime.

730^4//730^3//730^2//730^1//730^0 is prime, where // is the concatenation function.

211th Day of the Year.

211 = 121 + 11*(11-(1+2)) + 2*1*1
211 = 112 + 11*(11-2)
211 = (2+1+1)*((11+2)*(2+1+1))
211 = (11+2)*(12+(2+1+1)) + 2*1+1
211 = (21-1)*(11-1*1) + (21-(11-1))
211 = (21+1)*(11-2) + (11+2)
211 = (11-2)*(21+2*1) + (2+1+1)

211 is the 47th prime (47 is prime).

211 = 1 + 2*3*5*7

211 is concatenation of two primes 2 and 11.

211 = 67+71+73 (three consecutive primes)

211 = 3^5-2^5 = 14^2+14+1

There are 211 prime lucky numbers less than 10^4 (Note: 211 itself is a prime lucky number and 4 = 2+1+1).

There are 211 primes that appear on a 24-hour digital clock 00:00-23:59.

211 has its own website: www.211.org

July 31st (212th Day of the Non-Leap Year)

Jul 31st, 7/31.

731 = 713 + (7-1)*3
731 = 371 + 3*71 + 7*(7*3*1)
731 = 317 + 31*7 + (173+(1+7)*3)
731 = 137 + 13*7 + 317 + (7-1)*31
731 = 173 + 17*3 + 13^(7-(3!-1))*3
731 = 7*31 + 73*1 + (7*3*1)^(7-(3!-1))
731 = 1*37 + 13*7 + (7+3-1)*(73-3!)
731 = (7+31) + (73+1) + 371 + (7+1)*31
731 = (1+37) + (13+7) + 317 + (7-3*1)*(73+(3-1)^(7-3))
731 = (7+3+1)*(71-(3!-1)) + (7-(3!-1))
731 = (7*3+1)*(37-(7-3)) + (7-(3!-1))
731 = (7+3*1)*31 + (7-3!*1)
731 = (7*3*1)*(17*(3-1)) + (31-13-1)
731 = (73+1)*(7+3*1) - (7+3-1)
731 = (73-1)*(7+3*1) + (7+3+1)
731 = (7+31)*(7*3-(7-(3!-1))) + (13-(1+3))

731*(7*3*1) is a palindrome.

71, 31, and 73 are all prime, but 731 is composite.

If 731 = n, 7n7 and n7n are both prime (Meaning 77317 and 7317731 are prime).

If 731 = n, 101001000n is prime (Meaning 101001000731 is prime).

731*731! - 1 is prime.

If 731 = n, 2357n is prime (Meaning 2357731 is prime).

212th Day of the Year.

212 = 122 + (1+2+2)*(21-(2+1))
212 = 221 - (2+1)^2
212 = 2*12 + 21*2 + 22*1 + (2+1+2)^(2-1+2) - (2+1-2)
212 = (2*1*2)*(21+2^(2+1+2))
212 = (2+1+2)*(21*2) + 12/(2*(1+2))
212 = (21+2)*((2+1)^2) + (2+1+2)
212 = (2+12)*(21-2*(1+2)) + (2-1)*2
212 = (2*12)*(12-(1+2)) - (2*1*2)
212 = (21-2)*(12-(2-1)) + 2-(1-2)
212 = (12-2)*21 - 2*(1-2)

212^3+212+1 is prime.

212^10+212^9+212^8+212^7+212^6+212^5+212^4+212^3+212^2+212+1is prime.

212 and 212^2 contain two different digits.

3^212 - 212^3 is prime.
11^212-212^11 is prime.

DigitSum(212^2) and DigitProd(212^2) are both squares.

There are 212 primes of the form x^16+1 less than 10^61.

There are 212 4-digit numbers that are multiples of their digit reversal.

212^2 + 213^2 + 214^2 + 215^2 + 216^2 + 217^2 is prime.

212^2//212^1//212^0 is prime (// denotes the concatenation function).

212, 213, 214, 215, 216, 217, 218 and 219 all have the same number of distinct prime divisors.

August 1st (213th Day of the Non-Leap Year)

Aug 1st, 8/1.

81 = (8+1)^2

81 = 8*8+(8^(8/8)+8^(8/8))+1 (Eight 8's and One 1).

2^81 (25 digits) does not contain a 0.

81 is the smallest number of the form (k!+9)/9 (k = 6).

81^3-81-1 is prime.

81 divides 2^81+1.

2^81-81^2 is prime.

1/81 = .012345679012345679012... (missing the number 8).

213th Day of the Year.

213 = 231 - 2*(3^(2*1))

213 = 132 + (3^2)*(21/3-(2-1-3))

213 = 123 + 3*(21+3^2)

213 = 321 - 12*(2!+1!+3!)

213 = 312 - 3*(31+2)

213 = (2+1+3)*((21/3)*(2+1*3)) + (2-1)*3

213 = (21+3)*(2^3+1) - (21/(2^3-1))

213 = (21-3)*(2*3!) + 3*(1-2)

213 = (2+13)*(12+1*2) + 3!-2!-1!

213 = (13-2)*(21-2*1) + (2-1+3)

If 213 = n, 2n3n5n7n11n13 is prime (Meaning, 22133213521372131121313 is prime).

August 2nd (214th Day of the Non-Leap Year)

Aug 2nd, 8/2.

821, 823, 827, and 829 are all prime.

82 = 3^4+1 (3 is prime) and 82^4+1 is prime.

Nuclei with either 82 protons or neutrons are more stable against nuclear decay.

82^2+83^2 is prime.

82^4-82-1 is prime.

There are 82 ways to place 4 nonattacking queens on a 5 by 5 board.

The code *82 unblocks your caller ID for phones that block anonymous calls.

214th Day of the Year.

214 = 241 - (2+1)^(4-1)
214 = 412 - 142 - ((2-1)*4)*14
214 = 421 - 241 + (21-4)*(4/2)
214 = 142 + (2*1*4)*((4-1)^(2*1))
214 = 124 + (2*4+1)*(2*(1+4))
214 = (2+14)+(21+4)+(4+12)+(41+2)+(42+1)+(1+24)+ 2*(24-1)
214 = 21*4 + 41*2 + 4*12
214 = (21+4)*(2*1*4) + 21-(2*4-1)
214 = (21-4)*((2+1)*4)
214 = (2+14)*(21-2*1*4)+((2*1)+4)
214 = (14-2)*(21-(2+1)) - (4/(1*2))
214 = (2*1*4)*((4-1)^(2+1)) - 24/(4*(1+2))
214 = (2+1+4)*(24+(2+4)) + (2-1)*4

214!! - 1 is prime.

The sum of the first 214 primes is prime.

If 214 = n, 2n3n5n7n11n13 is prime (Meaning, 22143214521472141121413 is prime).

214^2 + 215^2 + 216^2 + 217^2 + 218^2 + 219^2 is prime.

214 and 2^214 both end in a 4.

August 3rd (215th Day of the Non-Leap Year)

Aug 3rd, 8/3.

83^3+83+1 is prime.

83^4+83^3+83^2+83+1 is prime.

83 is the 23rd prime number (Note: 23 is also prime).

83 = 23+29+31 (three consecutive primes).

83 = 11+13+17+19+23 (5 consecutive primes).

(Note that both equations have the 23 and 83 is the 23rd prime).

When a man turns 83 years old, he may celebrate a second bar mitzvah.

TI-83 is a type of calculator.

The smallest prime with a digit sum of 83 is 3999998999.

There are 83 9-digit squares with all 9 digits being different.

83^2 = 6889 is a strobogrammatic number.

83 is the only prime of the form p^4+2 for prime p.

The average of all primes up to 83 is 38.

2^83 (25 digits) does not contain a '2' (83 is the largest known prime exponent with this property).

215th Day of the Year.

215 = 251 - (5+1)^2

215 = 152 + (5*2-1)*(2+5*1)

215 = 125 + 5*(21-(2+1))

215 = 512 - 152 - 125 - 5*(5-2+1)

215 = 521 - 251 - (5+2*1)*(15-2)

215 = (21+5) + (2+15) + (5+12) + (51+2) + (25+1) + (1+52) + (21+2*1)

215 = (2*15) + (21*5) + (5*12) + 5*(5-2+1)

215 = (2+1+5)*(25+1) + (2+5*1)

215 = (2*1*5)*21 + (2-1)*5

215 = (2+15)*((5-1)*(2+1)) + (15-(5-1))

215 = (21-5)*(15-2) + 21/(2+1)

215 = 5*43 (decreasing digits)

215^2+215+1 is prime.

215^16 + 216^16 is prime.

215^2 + 4 is prime.

215^3 is the largest number whose cube has 7 digits.

The 215th and 216th primes are twin primes.

215!! + 2^7 and 215!! + 2^8 are primes (Note the base of 2 and 7+8=15). This is the only known number with this property.

215^2 + 215 +/- 1 are twin primes.

215, 217, 219, 221 are semiprimes.

There are 215 solutions to 1 = 1/a + 1/b + 1/c + 1/d for a,b,c,d positive integers (rearrangements included).

215 is the largest known number with a 7 in its aliquot sequence.

August 4th (216th Day of the Non-Leap Year)

Aug 4th, 8/4.

84 = 1 + (1+2) + (1+2+3) + (1+2+3+4) + (1+2+3+4+5) + (1+2+3+4+5+6) + (1+2+3+4+5+6+7)

84*84 = 7056

84*48 = 4032 (Note the similarities between these equations...)

84^2+1 is prime.

84 = 1^2+3^2+5^2+7^2

There is no prime between 840 and 849.

84^2+85^2 is prime.

84^2-84-1 is prime.

Los Angeles Lakers Magic Johnson holds the record for most assists (84) in a 6-game NBA final series in 1985.

There's a town in Pennsylvania, USA called Eighty Four.

It takes 84 years for Uranus to orbit the sun once.

216th Day of the Year.

216 = 6^(1+2)

216 = 126 + 6*(21-6)

216 = 162 + 6*(2+1+6)

216 = 261 - (2+1+6)*(6-(2-1))

216 = 612 - 261 - (6-1)*(21+6)

216 = 621 - 162 - 126 - (2+1+6)*(2*6+1)

216 = 2*16 + 21*6 + 2*1*6 + 21*(6-1-2)

216 = 6*12 + 61*2 + (6+1+2) + (1+6*2)

216 = (2+16) + (21+6) + (6+12) + (61+2) + (26+1) + (1+62)

216 = (2+1+6)*(6*(2-1))*(6-2*1)

216 = (2*1*6)*(2+16)

216 = (21-6)*(16-2) + 6*(2-1)

216 = (21+6)*(2*1+6)

216 = (2+16)*(21-(2+1+6))

216 = (3!)^3

216 is the smallest number whose cube has 8 digits (Note: 216 and 8 are both cubes).

The sum of the first 216 primes is prime.

216 is divisible by 2*1*6, 2+1+6, and 2,1, and 6.

216 = 4*6*9 (first 3 semiprimes).

3^216 +/- 32 are primes.

3^2 + 4^2 = 5^2 = 25.

3^3 + 4^3 + 5^3 = 6^3 = 216.

August 5th (217th Day of the Non-Leap Year)

Aug 5th, 8/5.

85^2+86^2 is prime.

There are 85 primes less than 21^2.

85^4+85^3+85^2+85+1 is prime.

85 and 58 both have 2 prime factors.

85 = 17*5 and 8+5 = 1+7+5.

85 = 4^3+4^2+4+1.

90*85+37 and 90*85+73 are both prime.

There are 85 primes between 43000 and 44000.

(This is the lowest number of primes between n*1000 and (n+1)*1000 until n = 64.)

85 = 125 base 8 (Note: 125 is 5^3 and 8 is 2^3. Further 5+3 = 8 and 2+3 = 5).

217th Day of the Year.

217 = 172 + (7+2)*(7-2)

217 = 127 + (1+2+7)*(12-(1+2))

217 = 271 - (21-(2+1))*(21/7)

217 = 721 - 271 - 21*7 - (72+7*2)

217 = 712 - 172 - 127 - (2*1*7)^(17-(2*7+1))

217 = 2*17 + 21*7 + (7-1)^2

217 = 71*2 + 7*12 - (7+1*2)

217 = (21+7) + (2+17) + (71+2) + (7+12) + (17-(7-2-1))*(7-2+1)

217 = (2+1+7)*(27-(7-2)) - (12/(7-2-1))

217 = (2*1*7)*(17-2) + (2-1)*7

217 = (21+7)*(7+2-1) - (17-(2+1+7))

217 = (2+17)*(12-(2-1)) + 2*(7-1-2)

217 = 7*31

217*(7+31) = 8246 (contains one of each nonzero even digit).

217^2+4 is prime.

There are 217 0's in all the four digit primes.

There are 217 squares with 5 digits.

August 6th (218th Day of the Non-Leap Year)

Aug 6th, 8/6.

2^86 (26 digits) contains no '0' (This is conjectured to be the highest exponent with such property).

86^2-86-1 is prime.

86^3-86-1 is prime.

86^10+86^9+86^8+86^7+86^6+86^5+86^4+86^3+86^2+86+1 is prime.

86^23+86^21+86^19+86^17+86^15+86^13+86^11+86^9+86^7+86^5+86^3+86+1 is prime.

There are 86 primes between 45000 and 46000.

There are 86 metals in the periodic table.

218th Day of the Year.

218 = 128 + (81+(8+1))

218 = 182 + (18+2) + 1*8*2

218 = 281 - (8+1)*(8-1)

218 = 812 - 281 - 21*8 - (1!+(8/2)!+(8-2-1)!)

218 = 821 - 128 - 182 - 81*2 - (2+1+8)^2 - (2*1+8)

218 = 21*8 + 2*18 + (21-(8-1))

218 = 8*12 + 81*2 - (8*(2-1))*(8-2-1)

218 = (2+18) + (21+8) + (8+12) + (81+2) + (8-2*1)*(8+2+1)

218 = (2+1+8)*((8/2)*(8-2-1)) - (28/(8*2-2*1))

218 = (2*1*8)*(2*(8-1)) - (8-1*2)

218 = (21+8)*(8-2+1) + (2+1)*(8/2+1)

218 = (21-8)*(28-(2+1+8)) - (8/2-1)

218^16 + 219^16 is prime.

218^2 + 218 +/- 1 are twin primes.

218^5 - 217^5 is prime.

218^5 - 218 - 1 is prime.

218^218 contains the digits '0123456789' consecutively in some order.

August 7th (219th Day of the Non-Leap Year)

Aug 7th, 8/7.

87^3+87+1 is prime.

87^2+88^2 is prime.

87^6+87^5+87^4+87^3+87^2+87+1 is prime.

87^2-87-1 is prime.

1/5+2/4+3/3+4/2+5/1 has 87 on the numerator.

87^2+4 is prime.

87 is 322 in base 5 and 223 in base 6.

In the National Hockey League, Wayne Gretzky scored 87 goals with the Edmonton Oilers in 1983-84.

There were 87 years between the signing of the Declaration of Independence and the Battle of Gettysburg (Thus, Abraham Lincoln's famous words were "Four score and seven years ago...").

219th Day of the Year.

219 = 129 + (92-2*1)

219 = 192 + (2+1)*9

219 = 291 - (21-(2+1))*(12/(1+2))

219 = 912 - 291 - 129 - 192 - 9^(2*1)

219 = 921 - 192 - 291 - 129 - 9*(9+2-1)

219 = 2*19 + 21*9 - (9-2+1)

219 = 9*12 + 91*2 - (91-(29-(9*(2-1))))

219 = (2+19) + (21+9) + (9+12) + (91+2) + 9*(9-1-2)

219 = (2+1+9)*(2*1*9) + (9/(1+2))

219 = (2+19)*(9+2-1) + (2-1)*9

219 = (21+9)*(9-2*1) + (21-(2+1+9))

219 = (91+2)*(9/(2+1)) - (9-2-1)*(9+2-1)

219 = (91-2)*(21/(9-2)) - (21-9)*((9-1)/2)

219^3 has 3 consecutive numbers in its decimal representation.

219^3 + 219 + 1 is prime.

There are 219 primes less than 37^2 (Note: 37 is prime).

219^4 + 219 + 1 is prime.

2^219 - 219^2 is prime.

There are more primes between 219 and 2*219 than 219 and 219^2.

There are 219 3-dimensional space groups.

If 219 = n, 2n3n5n7n11 is prime (Meaning 221932195219721911 is prime).

219^32 + 220^32 is prime.

August 8th (220th Day of the Non-Leap Year)

Aug 8th, 8/8.

88^4+1 is prime.

88^2 only contains 2 different digits.

You need 88 people in a room for a 50% chance that 3 people have the same birthday.

It takes around 88 days for Mercury to complete an orbit around the Sun.

A standard playing card is 88 mm.

There are 88 keys on a standard piano.

220th Day of the Year.

220 = 202 + (20-2)

220 = 22*(20/2)

220 = (20-2)*(20-(2^(2+2^0))) + 2*2+0

220 = (2+2+0)*(22+(20+(22-(2+2^0)^2)))

220 = (2*20)*(22/(2+2+0))

220 = 2*2*5*11

(2+2+5+11) = 20 (substring of 220)

220*202 = 44440

220 = 1 + (1+2) + (1+2+3) + ... + (1+2+3+4+5+6+7+8+9+10)

220^4 + 1 is prime.

220^6 + 220^5 + 220^4 + 220^3 + 220^2 + 220 + 1 is prime.

220^8 + 221^8 is prime.

The maximum difference between two consecutive primes less than 10^8 is 220 (Which primes?).

If 220 = n, n0123456789 is prime (Meaning 2200123456789 is prime).

The aliquot sequence for 220 does not terminate in 1.

220 = 47+53+59+61 (four consecutive primes).

August 9th (221st Day of the Non-Leap Year)

Aug 9th, 8/9.

2^89-1 is prime.

89 = (8+9)+(8*9)

89^2+89+1 is prime.

There is no prime between 890 and 899.

89^6+89^5+89^4+89^3+89^2+89+1 is prime.

89^16+90^16 is prime.

2*3*5*7*11*...*83*89 - 1 is prime.

89+98 = 187

187+781 = 968

...

After 24 iterations, the result will be a palindrome. No number < 10000 has this many iterations.

221st Day of the Year.

221 = 122 + ((2+1)^2)*(22/(2/2+1))

221 = 212 + (12-(1+2))

221 = (2*21) + (22*1) + (12*2) + 122 + (22/(2*1))

221 = (2+2+1)*(22*(2/2+1)) + (2/(2/1))

221 = (2*2*1)*(22+(2+1)*(12-(2-1))) + (2-2+1)

221 = (22-1)*(12-2) + (22/(2*1))

221 = (22+1)*((2+1)^2) + (12+2)

221 = (2+21) + (22+1) + (12+2) + (1+22) + 122 + (2+2)^(2/2+1)

221^3+221+1 is prime.

221^6 + 221^5 + 221^4 + 221^3 + 221^2 + 221 + 1 is prime.

221^8 + 222^8 is prime.

+/- 1 +/- 2 +/- 3 +/- ... +/- 13 = 1 has 221 solutions.

1111...11113 is prime (221 digits).

221^2 + 222^2 + 223^2 + 224^2 + 225^2 + 226^2 is prime.

There are 221 non-attacking knights on a 21 by 21 board.

If 221 = n, n0123456789 is prime (Meaning, 2210123456789 is prime).

221 = 37+41+43+47+53 (five consecutive primes).

221 = 11+13+17+19+23+29+31+37+41 (nine consecutive primes).

If you're dealt 2 cards from a single deck of 52 cards, you have a 1 in 221 chance both are the same face card.

August 10th (222nd Day of the Non-Leap Year)

Aug 10th, 8/10.

810 = 801 + (8+0+1)

810 = 108 + 10*8 + (10-8)*81 + 10*(81-(8-1)*(10/(1+8^0)))

810 = 180 + 1*80 + 10*(81-(18+1*8))

810 = 81*(8-1) + (10-(8-1))^(10/(1+8^0))

810 = 81*(8+1) + 81

810 = 81*(8+1+1^0)

810 = 18*(8-1) + (10+(18+1*8))*(18+18^0)

810 = 18*(8+1) + 8*81

810 = 81*18 - 801 + (8+1+0)*(18-1)

810 = (8+10) + (81+0) + (80+1) + 10*(8-1)*(8+1)

810 = (8+1+0)*(18+(10-8))

810 = (8+10)*(18*(10-8)+(8+1+0))

810^5 is the next power of 810 > 1 that has an 8.

810^61 + 810^59 + 810^57 + ... + 810^5 + 810^3 + 810^1 + 1 is prime.

810^2 - 810 - 1 is prime.

810 = 2*3*3*3*3*5. And 810^2 + 810^3 + 810^3 + 810^3 + 810^3 + 810^5 + 1 is prime.

222nd Day of the Year.

222 = (2+2+2)*(22+(2+2/2)*(2+2+2/2))
222 = (2*2^2)*((2^2)!+2^2) - 2*(2/2)!
222 = (2+2)!*(2+2/2)^2 + (2+2/2)!
222 = (22-2)*(22/2) + ((2+2)!-22)
222 = (2*22) + (2*2*2) + (22*2) + (22+2) + (2+22) + (2+2+2) + ((2+2/2)^2)*(2^2*2)
222 = (2^2^2)*(22-2*2*2) - (22/(2*2^2+(2+2/2)))

222 = 2*3*37
222*(2+3+(3+7)) = 3330 (another rep digit, excluding the 0).

222 = (3!)^3 + (2!)^2 + (1!)^1 + (0!)^0.

222^4 + 222^3 + 222^2 + 222 + 1 is prime.

10^222 + 7 is prime (Note: 1+0+2+2+2 = 7).

If 222 = n, 2n3n5n7n11n13 is prime (Meaning, 22223222522272221122213 is prime).

222 is 22020 in base 3 (only nonzero digits are 222).

August 11th (223rd Day of the Non-Leap Year)

Aug 11th, 8/11.

811 = 181 + 1*81 + 18*1 + (8+1*1)*(81-(18+8/(1+1)))

811 = 118 + 1*18 + 11*8 +

811 = 81*11 - (81-1)

811 = (8+1+1)^(11-8) - (8*1+1)*(18+(11-8))

811 = (8*1+1)^(11-8) + (81+1)

811 = 81*18 - 818 + (8*1+1)*(18+1)

811 = (8+11) + (81+1) + (18+1) + 181 + (8+1+1)*(18+11*(11-8))

811 = (8+1+1)*81 + (11-(8+1+1))

811 = (81+1)*(8+1+1) - (8*1+1)

811 = 18*11 + 118 + 11*(8+1)*(11-(8-1-1))

811*118 don't contain any 1's.

1/811 repeat after 810 decimal digits (longest possible).

There are 811 primes between 210000 and 220000.

811 is the sum of three cubes.

2^811, 3^811, 5^811, 7^811, 11^811, 13^811, 17^811, and 19^811 all have even digit sums.

223rd Day of the Year.

223 = 232 - (2+3+2+(3-2/2))
223 = 322 - (22*(3^2/2))
223 = 2*23 + 22*3 + (32+(3+2))*(3*(2/2))
223 = 3*22 + 32*2 + 3*(32-2/2)
223 = (2+23) + (22+3) + (32+2) + (3+22) + 2*(23+(32+2))
223 = (2+2+3)*(2^(2+3)) - (2+2-3)
223 = (2*2*3)*(23-(2+3)) + (2+2+3)
223 = (2+2*3)*(22+2*3) + (3-2*2)
223 = (22-3)*(2*2*3) - (2^3-(3*(2/2)))
223 = (22+3)*(3^2) + 2*(2-3)
223 = (23-2)*(22/(3-2/2)) - (2+2*3)

223^2 = 49729 (concatenation of 7^2 and 27^2).

223^3 contains '567' and the numbers 5678901 only.

223 = -01+23+45+67+89

223^16 + 224^16 is prime.

223 = 71+73+79 (three consecutive primes).

223 = 19+23+29+31+37+41+43 (seven consecutive primes).

August 12th (224th Day of the Non-Leap Year)

Aug 12th, 8/12.

812 = 821 - (8+(2-1))

812 = 281 + 2*81 + (8+(2-1))*(18+(21+2*1))

812 = 218 + 21*8 + (8-1*2)*(82-(8+1+2))

812 = 128 + 12*8 + 21*28

812 = 182 + 1*82 + 18*2 + 2^(8+1)

812 = 8*12 + 81*2 + 8*1*2 + 281 + (2^8+1)

812 = 2*18 + 21*8 + (8*1*2)*(28+(2+8))

812 = (8+12) + (81+2) + (8+1+2) + 2*(2^8+(2+1)*(28+(8/2-1)))

812 = (2+18) + (21+8) + (8-1)*(81+28)

812 = (8+1+2)*(81-8*1) + (2-(1-8))

812 = (8*1+2)*81 + 8/(12-8)

812 = (8+12)*(28+12) + (8/2)*(2+1)

812 = (81+2)*(8+1*2) - (8+1)*2

812 = (81-2)*(8*1+2) + (28-(8-2))

812^2 does not contain an 8, 1, or 2.

812^10 + 812^9 + 812^8 + ... + 812^2 + 812^1 + 1 is prime.

812 is the smallest integer where 29*n/(29+n) is an integer.

4*812^2 +/- 3 are consecutive primes.

There are 812 primes between 42^pi and 43^pi.

If 812 = n, then 1n2n3n4n5n6n7n8n9 is prime (Meaning 181228123812481258126812781288129 is prime).

224th Day of the Year.

224 = 242 - (22-4)
224 = 422 - (4^2+2)*(2*2*4-(4+2/2))
224 = 2*24 + 22*4 + (42+2)*(4-2)
224 = 4*22 + 42*2 + (24+2)*(4-2)
224 = (2+24) + (22+4) + (4+22) + (42+2) + 2*(2*24+(4-2/2))
224 = (2+2+4)*(4*(2+(4+2/2)))
224 = (2*2*4)*(4+4*2+2)
224 = (22-4)*(4*(4-2/2)) + (2*2+4)
224 = (22+4)*(2+2+4) + 2*2*4
224 = (2+2*4)*(24-2) + 4*(2/2)

224 = 2*2*2*2*2*7
2+2+2+2+2+7 = 17 (also prime).

224^3 begins with the first four terms of the Fibonacci sequence (1123...)

224^2+1 is prime.

224^10 + 224^9 + 224^8 + 224^7 + 224^6 + 224^5 + 224^4 + 224^3 + 224^2 + 224 + 1 is prime.

2^224, 3^224, 5^224 all have even digit sums.

224^3 +/- 2 are both prime.

224^2 + 225^2 + 226^2 + 227^2 + 228^2 + 229^2 is prime.

4^224 - 2^224 - 1 is prime.

224 = 2^3 + 3^3 + 4^3 + 5^3

August 13th (225th Day of the Non-Leap Year)

Aug 13th, 8/13.

813 = 831 - 3*3!

813 = 138 + 13*8 + 1*38 + 13*(31+(8-(1-3)))

813 = 183 + 1*83 + (8*3-1)^(8/(1+3)) + 3*(8-(3-1))

813 = 381 + 3*81 + (8+13)*(13-(1+3))

813 = 318 + 31*8 + 13*(13+(8-(3-1)))

813 = 8*13 + 81*3 + 8*1*3 + 13*(31+3*1)

813 = 3*18 + 31*8 + (81-8*1)*(3!+1)

813 = (8+13) + (81+3) + (8+1+3)*(38+3*(8-1))

813 = (3+18) + (31+8) + 3*((3-1)^8-(13-8))

813 = (8+1+3)*(83-8*(3-1)) + (13-(1+3))

813 = (8*1+3)*(81-(8-1)) - (8-1-3!)

813 = (81+3)*(3^(8-3!)) + 3*(31-(8+1+3))

813 = (81-3)*(8*1+3) - 3*(18-3)

813 = (8+13)*38 + (18-3)

813^2 does not contain an 8, 1, or 3.

813 is of the form n^2 - n + 1 (n = 29, a prime) and 813^2 - 813 + 1 is also prime.

813 is a divisor of 99999.

There are the same number of primes in [813,2*813] and [813^2,814^2].

813 is a semiprime. 813^2 +/- 2 are also both semiprimes.

If 813 = n, n7n and 7n7 are prime (Meaning 8137813 and 78137 are prime).

The largest prime with 2^8 digits is 10^256 - 813.

The sum of the first 22 primes + 22 = 813.

813^e begins with 813.

225th Day of the Year.

225 = 252 - (5-2)^(5-2)
225 = 522 - 252 - 5*(2+2+5)
225 = (2*25) + (22*5) + 5*(22-(2+2+5))
225 = 5*22 + 52*2 + (5*2+2/2)
225 = (2+25) + (22+5) + (52+2) + (5+22) + (5*2)*(5+2+2)
225 = (2+2+5)*25
225 = (2*2*5)*(2*5 + 2/2) + 5*(2/2)
225 = (2+25)*(5*2-2) + (5-2)^2
225 = (22-5)*(5*2+(5-2)) + (5-2/2)
225 = (25-2)*(22-(2+2*5)) - ((22+(5-2))/5)

225 = 15^2 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3.

225 = (3!)^3 + (2!)^3 + (1!)^3.

225 = 01+23+45+67+89

225^3+225+1 is prime.

225^16 + 226^16 is prime.

(6+225!)/6 is prime. (Second known largest number with property).

225 is the only three digit square with all prime digits.

A Scrabble board has 225 squares.

August 14th (226th Day of the Non-Leap Year)

Aug 14th, 8/14.

814 = 841 - (18+(1+8))

814 = 481 + 4*81 + (14-(1+4))

814 = 418 + 41*8 + 4*(4!-(8-1))

814 = 148 + 1*48 + (14-8)*(8*14-(8+1))

814 = 184 + 18*4 + (8*4-1)*18

814 = 8*14 + 81*4 + 18*(8-1)*(4-1)

814 = 4*18 + 41*8 + 18*(4!-1)

814 = (8+14) + (81+4) + (8-1)*(8*1*4+(4-1)*(4!-1))

814 = (4+18) + (41+8) + 418 + (4!+1)*(8+1+4)

814 = (8+1+4)*(81-(14+(1+4))) + (4-1)!+(8/4)

814 = (8*1+4)*(81-14) + (8/4)*(4+1)

814 = (8+14)*(41-4*1)

814 = (81+4)*(8+1) + (8-1)^(8/4)

814 = (81-4)*(14-1*4) + 4*(8+4-1)

814*814 does not contain an 8, 1, or 4.

814^5 is the next power of 814 > 1 that contains an 8.

814^29 + 814^27 + 814^25 + ... + 814^3 + 814^1 + 1 is prime.

814^40 + 814^39 + 814^38 + ... + 814^2 + 814^1 + 1 is prime.

If 814 = n, then 1n2n3n4n5n6n7n8n9 is prime

(Meaning, 181428143814481458146814781488149 is prime).

226th Day of the Year.

226 = 262 - 6^(6-2-2)
226 = 622 - 262 - 2*(62+(6-2/2))
226 = (2+26) + (22+6) + (62+2) + (6+22) + 6*(26/2)
226 = 2*26 + 22*6 + 6*(6+2/2)
226 = 6*22 + 62*2 - 6*(6-2/2)
226 = (2+2+6)*((6+(6-2/2))*(6-2-2)) + 6*(2/2)
226 = (2*2*6)*((6/2)^2) + (2+2+6)
226 = (2+2*6)*(2^(6-2)) + (6-2-2)
226 = (22+6)*(2^(6/2)) + ((6-2)/2)
226 = (22-6)*(2+2*6 + (2/2)/(6+2)

226^5 - 225^5 is prime.

226^2 + 227^2 + 228^2 is prime.

226 = (3!)^3 + (2!)^3 + (1!)^3 + (0!)^3.

August 15th (227th Day of the Non-Leap Year)

Aug 15th, 8/15.

815 = 851 - (8+1)*(5-1)

815 = 581 + 58*1 + ((5-1)!-(8-5-1))*8

815 = 518 + 5*18 + (8+15)*(8+1)

815 = 185 + 1*85 + 5*(81+(8*5-(8+5)))

815 = 158 + 15*8 + 3*(158+(15+(1+5)))

815 = 8*15 + 81*5 + 5*58

815 = 5*18 + 51*8 + (158*(8-5-1)+1)

815 = (8+15) + (81+5) + (8+1+5) + (5-1)*(158+15)

815 = (5+18) + (51+8) + (8-5)*185 + (8-5-1)*(81+8*1)

815 = (8+1+5)*58 + (8-5*1)

815 = (8*1+5)*(81-18) - (8-(5-1))

815 = (8+15)*(5*(8-1)) + 5*(8-5-1)

815 = (81+5)*(15-(1+5)) + (8*5+1)

815 = (81-5)*(15-(5-1)) - (15+(1+5))

1^4*2^4*3^4*...*815^4 is divisible by 1^4+2^4+3^4+...+815^4.

227th Day of the Year.

227 = 272 - (7-2)*(7+2)
227 = 722 - 272 - (22-7)^2 + (7-2-2-2/2)
227 = 2*27 + 22*7 + (7*2+(7-2))
227 = 7*22 + 72*2 - (72-2/2)
227 = (2+27) + (22+7) + (7+22) + (72+2) + 22*(7-2-2)
227 = (2+2+7)*(7*(7-2-2)) - (7+2/2)/2
227 = (2*2*7)*(2/2+7) + (7-2-2)
227 = (7-2-2)*(72+(7-2)) - (7-2-2/2)
227 = (7-2/2)*(22+2^(2+2)) - (7-(2*2*2))
227 = (22-7)*(7*2+2/2) + (7-(7*2-(7+2)))
227 = (22+7)*(2/2+7) - (7*2-(7+2))

227*227 contains all 6 different digits.

227^4 + 227^3 + 227^2 + 227 + 1 is prime.

22/7 approximates the famous value of pi (3.14).

227 = 2*3*5*7 + (2+3+5+7)

227 is the smallest three digit prime where 2^227 - 227^2 is prime.

227 is the smallest three digit prime where deleting a digit results in a composite.

The sum of the first composites up to 27 is 227.

There are 227 composite days in a non leap year.

August 16th (228th Day of the Non-Leap Year)

Aug 16th, 8/16.

816 = 861 - (61-16)

816 = 168 + 16*8 + (8+6-1)*(8*(6-1))

816 = 186 + 1*86 + 16*(18+16)

816 = 618 + 6*18 + 6*(8+1+6)

816 = 681 + 68*1 + (68-1)

816 = 8*16 + 81*6 + (16/8)*(86+(8+1+6))

816 = 6*18 + 61*8 + (16-1*6)*(16+1*6)

816 = (8+16) + (81+6) + (8+1+6)*(8*6-1)

816 = (6+18) + (61+8) + (8-(6-1))*(186+(61-6*1))

816 = (8+1+6)*(6*(8+1)) + 18/(8-(6-1))

816 = (8*1+6)*(68-(16-1*6)) + 8/(8-6*1)

816 = (8+16)*(16+18)

816 = (81+6)*(16-(1+6)) + (8-(6-1))*(16-(6-1))

816 = (81-6)*(18-1*8) + 6*(16-(6-1))

816 = 2*2*2*2*3*17

816 +/- (2+2+2+2+3+17) are both near repdigits.

228th Day of the Year.

228 = 282 - 2*(28-2/2)
228 = 822 - 282 - (8-2)*(2+2)*(8+(8/2+2/2))
228 = (2+28) + (22+8) + (82+2) + (8+22) + (8+2/2)*(8-2)
228 = 2*28 + 22*8 - (8-2-2)
228 = 8*22 + 82*2 - 2*(2*28)
228 = (22+8)*(8-2/2) + (8-2)*(8/2-2/2)
228 = (22-8)*(2*8) + (8-2-2)
228 = (2+2+8)*(8*2+(8-2)/2)
228 = (2*2*8)*(8-2/2) - (2+2-8)
228 = (2+2*8)*(8*2-(8-2)/2) - (22-2*8)

228^4+1 is prime.

There are 228 primes less than 38^2.

There are 228 ways to place 2 non attacking kings on a 5 by 5 board.

There are 228 primes between 39 and 39^2.

There are 228 composites less than or equal to 17^2.

228 = 29+31+37+41+43+47 (6 consecutive primes).
228 = 7+11+13+17+19+23+29+31+37+41 (10 consecutive primes).

August 17th (229th Day of the Non-Leap Year)

Aug 17th, 8/17.

817 = 871 - (8*7*1-(8-(7-1)))

817 = 718 + (18-7)*(17-8)

817 = 781 - 78*1 + (7-1)*(18+(8-7*1))

817 = 178 + 17*8 + (17-8)*(8*1*7) - (8/(1+7))

817 = 187 + 18*7 + 8*((8+1)*7)

817 = 8*17 + 81*7 + (8*7*1+1)*(8-(7-1))

817 = 7*18 + 71*8 + (8*7-(8+7))*(17-7*(8-(7-1)))

817 = (8+17) + (81+7) + 8*(1+7)*(18-7)

817 = (7+18) + (71+8) + (17+(7-1))*((8+7+1)+(8*1+7))

817 = (8+1+7)*(17*(18/(7-1))) + (8-1-7)!

817 = (8*1+7)*(8+1)*(7-1) + (78-71)

817 = (8+17)*(17+18) - (8*7+(8-(7-1)))

817 = (81+7)*(8+1) + (8+17)

817 = (81-7)*(18-7) + (18/(7-1))

If 817 = n, 9n9 and n9n are both prime (Meaning 98179 and 8179817 are both prime).

817//815//813//...//7//5//3//1 is prime (// is the concatenation function).

817^k is conjectured to end in any number of the same digit for certain k-values.

Ex. 817^11 ends in 33.

817^17 ends in 777.

817^217 ends in 7777.

817^1217 ends in 77777.

...It is believed that 817^k will end in any number of 7s given the correct k.

229th Day of the Year.

229 = 292 - 9*(9-2)

229 = 922 - 92*2 - 9*22 - 292 - (9*2+2/2)

229 = 2*29 + 22*9 - (29-2)

229 = 22*(2+9) - (2+2+9)

229 = 22*(9-2) + (9-2-2)*(22-(9-2))

229 = 29*(9+2) - (92-2)

229 = 29*(9-2) + (22+(2+2))

229 = 92*(9-2) - 292 - (2*2*9+92-(9-2-2))

229 = 92*(9+2) - 292 - 29*9*2 + (2+29)

229 = (2+29) + (22+9) + (2+2+9) + (9+2)*(9-2)*2

229 = (9+22) + (92+2) + (9-2/2)*(9+2+2)

229 = (2+2+9)*(9*2-2/2) + (9-2/2)

229 = (2+2*9)*(2+9) + 9*2/2

229 = (2+29)*(9-2) + (2*(9-2-2/2))

229 = 4^4 - 3^3.

1/229 repeats after 228 digits (longest run possible).

229^10 + 229^9 + 229^8 + ... + 229^3 + 229^2 + 229^1 + 1 is prime.

229^4 - 229 - 1 is prime.

n! never ends in 229 zeros.

229^19 + 229^17 + 229^15 + ... + 229^5 + 229^3 + 229^1 + 1 is prime.

229^8 + 230^8 is prime.

The 229th Fibonacci number has 5 prime factors (smallest Fibonacci number to have 5 prime factors).

229 is the smallest prime such that 229 + 922 (1151) is also prime.

229 is the highest score in bowling if your ten frames each have to be a prime number.

The sum of the first 229 primes divides the product of the first 229 primes.

229 is prime. Square each digit.

4481 is also prime.

229 is prime. Cube each digit.

88729 is also prime.

August 18th (230th Day of the Non-Leap Year)

Aug 18th, 8/18.

818 = 881 - (8*8-1)

818 = 188 + 18*8 + (8-1-8/8)*81

818 = 81*(8-1) + 188 + (8-1)*(8+1)

818 = 81*(8+1) + (81+8)

818 = 18*(1+8) + (8+8)*((8+1*8)+(18+1*8))

818 = 81*18 - 881 + 188 + (81-(18+(1+8+8/8)))

818 = 8*18 + 81*8 + (8+18)

818 = (8+18) + (81+8) + (18+1)*(18*(1+8/8)+1)

818 = (8+1+8)^(1+8/8)*(18/(8-1-8/8)) + (18*(1+8/8)-8/(1+8/8))

818 = (8+18)*((8+8)*(1+8/8)-1) + (18-(8-1-8/8))

818 = (81+8)*(1+8) + (8+1+8)

818 = (81-8)*(18-(8-1)) + (8-1+8)

818^5 is the next power of 818 > 1 that has an 8.

818*(8*1*8) is comprised of only prime digits.

818 is the smallest semiprime with exactly 2 8's.

818999 and 819001 are twin primes.

818^3 does not contain an 8 or 1.

818 is the smallest semiprime such that there are 4 primes between it and the next semiprime.

818! is the smallest factorial less that (41!)^41.

230th Day of the Year.

230 = 203 + 3^(2+2^0)

230 = 320 - 3^2*(30/3)

230 = 302 - (2+3)!-2*(23+23^0)

230 = 2*30 + 3*20 + (2+3+0)*(23-23^0)

230 = 23*(2+3)*2

230 = 32*(2+3) + 2*(32+3)

230 = 23*2*3 + (3+2^0)*23

230 = 32*3*2 + (23+30/2)

230 = (2+3+0)*(23*2)

230 = (2+30)*(2^3-2^0) + 2*3

230^2 + 1 is prime.

2, 3, 2+3, 2+0, 3+0 are all prime.

4000...0001 is prime (230 digits).

230^4 - 230 - 1 is prime.

1 + 1/2 + 1/3 + ... + 1/230 has a prime on the numerator.

230^5 - 229^5 is prime.

230^6 + 230 + 1 is prime.

11^230 - 230^11 is prime.

230^9 + 230^7 + 230^5 + 230^3 + 230^1 + 1 is prime.

If 230 = n, n0123456789 is prime (Meaning 2300123456789 is prime).

230999 and 231000 are twin primes.

August 19th (231st Day of the Non-Leap Year)

Aug 19th, 8/19.

819 = 891 - 8*9*1

819 = 981 - 98*1 - 8^(9-(8-1))

819 = 918 - 9*18 + (8-1)*9

819 = 198 + 19*8 + 8*1*9 + ((91+8)*(8/(9-(8-1)))+1)

819 = 189 + 1*89 + 198 + (8-1)^(9-(8-1)-1)

819 = 8*19 + 81*9 - (81-19)

819 = 9*18 + 91*8 - (89-18)

819 = (8+19) + (81+9) + 18*(81-(8-1)*(8-(9-(8-1))))

819 = (9+18) + (91+8) + 9*(8-1)*(19-8)

819 = (8+1+9)*(18+(19+8)) + 81/9

819 = (8*1+9)*(8*(8-(9-(8-1)))) + (9-(8-1-1))

819 = (8+19)*(19+(19-8)) + 9*(9-8)

819 = (81+9)*9 + (8+1)^(9-8)

819^5 is the next power of 819 > 1 that has an 8.

819 of the first 1000 primes do not contain a 0.

819 divides 999999.

There are 819 primes between 220000 and 230000.

819 numbers between 1 and 1000 don't have a 0.

819^5 +/- 2 are consecutive primes.

8*819 +/- 1 are twin primes.

819 = 4^5 - 4^4 + 4^3 - 4^2 + 4^1 - 1.

If 819 = n, 1n1 and n1n are both prime (Meaning 18191 and 8191819 are both prime).

231st Day of the Year.

231 = 213 + (21-3)

231 = 321 - 3*21 - 3^(2+1)

231 = 312 - 31*2 - (21-2*1)

231 = 132 + 1*32 + (31+3!^2)

231 = 123 + 12*3 + 3!*12

231 = 2*31 + 23*1 + (123+23)

231 = 1*32 + 13*2 + 132 + (32+3^2)

231 = (2+31) + (23+1) + 3!*(23+2*3)

231 = (1+32) + (13+2) + 3*(2*31-1)

231 = (2+3+1)*(32+3*2) + 3*(2-1)

231 = (2*3+1)*(2+31)

231 = (2+3*1)*(23*2) + 3/(2+1)

231 = (23+1)*(3^2+1) - 1*3^2

231 = (23-1)*(21/2)

231 = 3*7*11, 3+7+11 = 21.

231 = 1+2+3+...+21.

There are 231 distinct ways to break up the number 16 by integers (1+15, 1+1+14, ...).

There are five known sets of 9 distinct odd numbers whose sum of reciprocals is 1.

One of them is {3, 5, 7, 9, 11, 15, 35, 45, 231}.

Another is {3, 5, 7, 9, 11, 15, 21, 231, 315}.

231^8 + 232^8 is prime.

231^16 + 232^16 is prime.

231^4 - 231 - 1 is prime.

231 is a divisor of 111111.

There are 231 numbers < 10^4 with 7 prime factors.

231^2 + 231 +/- 1 are twin primes.

2309 and 2311 are twin primes.

4^231 - 3^231 - 2^231 is prime.

231^35 + 231^33 + 231^31 + ... + 231^5 + 231^3 + 231 + 1 is prime.

231^6 +/- 2 are consecutive primes.

There are 231 cubic inches in a U.S. liquid gallon.

August 20th (232nd Day of the Non-Leap Year)

Aug 20th, 8/20.

820 = 802 + (20-2)

820 = 280 + 2*80 + (8+2)*(28+(2+8))

820 = 20*(8+2) + 280 + 208 + (8+2+0!)*(20-8)

820 = 28*(2+8) + (8+2)*(28*2-2)

820 = 28*(2*8) + (20-8)*(28+(8/2-0!))

820 = 82*(8*2) - (20-8)*(20+(2+0!)*(8-0!))

820 = 82*28 - 802 - 280 - 208 - (8-2)*(20+(8+2+0!))

820 = (8+20) + (82+0!) + 280 + (8+2+0!)*(28+8+2+0!)

820 = (8+2+0)*82

820 = (8+20)^2 + (8-2)^2

820 = (20-8)*(82-2*(8-0!)) + (8/2)*0!

820 = 82*20 - 802 - (20-2*0!)

8+2+0 = 10

820 = 2*2*5*41 and 2+2+5+41=50, a multiple of 10.

820 = 1 + 2 + 3 + ... + 40.

820 = 9^3 + 9^2 + 9^1 + 1.

821 is the smallest prime factor of 820^3 +3^820.

232nd Day of the Year.

232 = 223 + (2^3+2/2)

232 = 322 - 32*2 - 2*(2^3+2+3)

232 = 23*(2+3) + 3^2*(23-2*(3+2))

232 = 32*(3+2) + (2*3*2)*3!

232 = 32*(3*2) + (32+2^3)

232 = 23*(2*3) + 2*(23*2+1)

232 = 23*32 - 322 - 2*3^2*23

232 = 3^2*2^3 + 32*(3+2)

232 = 2*32 + 23*2 + (2+3^2)^2+2/2

232 = (2+32) + (23+2) + (2+3+2) + 2*(2^3*3^2+(3^2+2))

232 = (2+3+2)*32 + (2*3+2)

232 = (2*3+2)*(23+2*3)

232 = (23+2)*3^2 + (3!+2/2)

232 = (23-2)*(2+3^2) + (2-(3-2))

There is no prime between 2320 and 2329.

232 = 4^4 - 4!

August 21st (233rd Day of the Non-Leap Year)

Aug 21st, 8/21.

821 = 812 + (12-(1+2))

821 = 281 + 2*81 + 18*21

821 = 218 + 21*8 + (8*2-1)*(8+21)

821 = 128 + 12*8 + (1+2)*(182+(8*2+1))

821 = 182 + 1*82 + (8+2+1)*(2*(8-1))*(21+18)

821 = 8*21 + 82*1 + 8*2*1 + (8-2-1)*(21+8*2*1)*(21/(8-1))

821 = 1*28 + 12*8 + (8*2+1)*(21+(8-2-1)*(8/2))

821 = (8+21) + (82+1) + (8+2+1) + 2*(281+(81-(21-8)))

821 = (1+28) + (12+8) + (8/2)*(182+(1+8+2))

821 = (8+2+1)*(82-8) + (8-(2-1))

821 = (8*2+1)*(28+21) - (8/2)*(2+1)

821 = (8+2*1)*82 + (82-81)

821 = (8+21)*28 + 81/(8+1)

821 = (82+1)*(8+1) + (82-8)

821 = (82-1)*(8+2) + (8+2+1)

821, 823, 827 and 829 are all prime.

1/821 has 820 decimal digits until it repeats (longest possible).

821^39 + 821^37 + 821^35 + ... + 821^5 + 821^3 + 821^1 + 1 is prime.

If 821 = n, 1n1 and n1n are both prime (Meaning 18211 and 8211821 are both prime).

If 821 = n, n0123456789 is prime (Meaning 8210123456789 is prime).

233rd Day of the Year.

233 = 332 - 3*33

233 = 323 - 3*(23+3^2-2)

233 = (3*3)^2 + 2^3*(23-(3!-2))

233 = 2^(3*3) + (3!+2!+(3+2))^2

233 = 23*3 + 2*33 + 2*(3*3-2)^2

233 = 3*32 + 33*2 + (23+2*3*2^3)

233 = 2^3*3^2 + 23*(3*3-2)

233 = (2+33) + (23+3) + (2*23-3)*(3!-2)

233 = (2+3+3)*(23+2*3) + (2-3/3)

233 = (23+3)*3^2 - (2-3/3)

233 = (23-3)*(2*3!) - (3^2-2)

233 = (2+33)*3! + 23

233 +- 2*3*3 use the same digits.

233 = (3!)^3 + (2!)^4 + (1!)^5.

1/233 has 232 decimal digits before it repeats.

233^11 + 233^9 + 233^7 + 233^5 + 233^3 + 233^1 + 1 is prime.

233 is a Fibonacci prime and 2+3+3 = 8 is a Fibonacci number.

233 is the 13th Fibonacci number. 13 is the 7th Fibonacci number. (7, 13, and 233 are all prime.)

233 is the only known Fibonacci prime whose digits are also Fibonacci primes.

Ray Bradbury's book "Fahrenheit 451" would've been called "Celsius 233" if he used Celsius.

233+(2+3+3), 233+2*3*3, 233+2^3*3, 233+(2+3)*3!, and 233+2*3*3!,  are consecutive primes.

233 is the largest known Fibonacci prime with only two different digits.

August 22nd (234th Day of the Non-Leap Year)

Aug 22nd, 8/22.

822 = 282 + 2*82 + 8*(22+(8-2-2/2)^2)

822 = 228 + 22*8 + 22*(28-(8+2/2))

822 = 82*(8-2) + (8+2)*(28+8/2+2/2)

822 = 28*(2+8) + 2*(2+8) + (28+2/2)*(8*2+2)

822 = 2*8^2 + 2*(282+((2+8)/2)*(22-8-2/2)

822 = 2*2^8 + (8+2)*(22+(8+2/2))

822 = 8*22 + 82*2 + (282+(8+2)*(22-2))

822 = (8+22) + (82+2) + (8+2+2)*(28*2+(8-2-2-2/2))

822 = (8+2+2)*(8^2+8/2) + (2+2)!/(2+2)

822 = (8*2+2)*(28+2*8) + (28+2)

822 = (8^2+2)*(8+2+2) + (8-2)*(2+2+2/2)

822 = (82+2)*(8+2) - (8*2+2)

822 = (82-2)*(8+2) + (8*2+(8-2))

822*2^822-1 is prime.

234th Day of the Year.

234 = 243 - (2^4-4-3)

234 = 342 - 34*2 - 2^3*(2+3)

234 = 324 - 3*24 - 3*(2+4)

234 = 423 - 42*3 - 3^2*(3+4)

234 = 432 - 4*32 - (2*3+4)*(3+4)

234 = 2*3^4 + 2*3*3*4

234 = 4*3^2 + (4*2+3)*(3*(2+4))

234 = 2*34 + 23*4 + 2*(32+(3+2))

234 = 4*32 + 43*2 + (2+3)*4

234 = (2+34) + (23+4) + 3^2*(23-4)

234 = (4+32) + (43+2) + 3^2*(23-2*3)

234 = (2+3+4)*(24+(4-2))

234 = (2*3+4)*23 + 4*(3-2)

234 = (2+3*4)*(4!-2^3) + (2*3+4)

234 = (2^3+4)*(23-4) + (3!-(4-2))

234 = (2+3^4)*3 - (23-2^3)

234 + (2+3+4) = 3^5.

234 - (2+3+4) = 15^2.

(both perfect powers)

234 = (3!)^3 + (2!)^4 + (1!)^5 + (0!)^6.

234^128 + 1 is prime.

234^96 + 234^95 + 234^94 + ... + 234^3 + 234^2 + 234 + 1 is prime.

August 23rd (235th Day of the Non-Leap Year)

Aug 23rd, 8/23.

823 = 832 - (23-2*3-8)

823 = 382 + 3*82 + (2+3)*(28+3+8)

823 = 328 + 3*28 + (8*2*3)*(3*(8+3)^2)

823 = 238 + 23*8 + (28-3)*(8*2)+(3^2-8)

823 = 283 + 2*83 + 2*(8+3)*(8+3^2)

823 = 8*23 + 82*3 + 3*((8+3)*(8*2-3)-(8/2)*3)

823 = 3*28 + 32*8 + 23*(8*2+(2+3))

823 = (8+23) + (82+3) + (8-(3-2))*(8*2*3 + 28*2-3)

823 = (3+28) + (32+8) + 8*2*(23+8*3)

823 = (8+2+3)*8^2 - 3^2

823 = (8*2+3)*(38+(8-3)) + (38-32)

823 = (8+2*3)*(32+3^(8-3-2))

823 = (8+23)*(28-2) + (8+3^2)

823 = (82+3)*3^2 + (23+38-3)

823 = (82-3)*(8+3) - 23*(8-3*2)

1/823 repeats after 822 decimal digits (longest run possible).

823 divides any Fibonacci sequence.

823*823*823 does not contain an 8, 2, or 3.

823 = 12345/(1+2+3+4+5).

235th Day of the Year.

235 = 253 - (23-5)

235 = 325 - 3^2*(2+3+5)

235 = 352 - 35*2 - (32+3*5)

235 = 532 - 5*32 - 53*2 - (32-(3-2))

235 = 523 - 5*23 - 52*3 - (2+3*5)

235 = 2*35 + 23*5 + (2+3)*5*2

235 = 5*32 + 53*2 - (23+2^3)

235 = (2+35) + (23+5) + (2+3+5)*(2+3*5)

235 = (5+32) + (53+2) + (5*3-2)*(2*3+5)

235 = (2+3+5)*23 + 35/(2+5)

235 = (2*3+5)*(25-3) - 35/(2+3)

235 = (2+3*5)*(23-3^2) - (2^3-5)

235 = (23+5)*(3+5) + (2*3+5)

235 = (23-5)*(5*3-2) + 5/(3+2)

235 = (2+35)*3! + (5*3-2)

235^6 + 235^5 + 235^4 + 235^3 + 235^2 + 235^1 + 1 is prime.

235! - 235th prime is prime.

235 is the smallest number such that 235^2 has a proper subset of the digits {2,3,5}.

August 24th (236th Day of the Non-Leap Year)

Aug 24th, 8/24.

824 = 842 - (2+4)*(24/8)

824 = 482 + 4*82 + (8+2+4)

824 = 428 + 42*8 + (8+2)*(2+4)

824 = 248 + 24*8 + (4+2)*8^2

824 = 284 + 28*4 + 428

824 = 8*24 + 82*4 + 2^4*(24-(4+8/(2*4)))

824 = 4*28 + 42*8 + 2*4*(48-8/(4*2))

824 = (8+24) + (82+4) + 2*(284+(24/8)*(4!-8/(2*4)))

824 = (4+28) + (42+8) + 2*(248+(24/8)*(42-8/(4*2)))

824 = (8+2+4)*(42+4^2) + 4*(24/8)

824 = (8*2+4)*(42-(8/4)) + (8-2)*4

824 = (8+2*4)*(24+(4!+(24/8)))

824 = (82+4)*(24/8)^2 + (48+2)

824 = (82-4)*(8+2) + (42+8/4)

824^1024 + 1 is prime.

824^27 + 824^25 + 824^23 + ... + 824^5 + 824^3 + 824^1 + 1 is prime.

236th Day of the Year.

236 = 263 - (6/2)^3

236 = 362 - 3*62 + 2*(2+3)*6

236 = 326 - 3*26 - (2*3+6)

236 = 623 - 6*23 - 3*(63+(2+3*6))

236 = 632 - 6*32 - (6-2)*3*(36/2-(6/(3*2)))

236 = 2*36 + 23*6 + 26*(3-2)

236 = 6*32 + 63*2 - (62+(2+3*6))

236 = (2+36) + (23+6) + (6*3-(3+2))^2

236 = (6+32) + (63+2) + (6*3+6/(3*2))*(63/(6+3))

236 = (2+3+6)*(26-3-2) + (6/2+2)

236 = (2*3+6)*(2+3*6) - 36/(3+6)

236 = (2+36)*6 + (6*2-(6-2))

236 = (23+6)*2^3 + 36/(3+6)

236 = (23-6)*(2^3+6) - (2^3-6)

236^2 + 1 is prime.

The 236th and 237th primes are twin primes.

236^2 + 236 +/- 1 are twin primes.

1999...999 (236 9's) is prime.

August 25th (237th Day of the Non-Leap Year)

Aug 25th, 8/25.

825 = 852 - (8-5)^(5-2)

825 = 582 + 5*82 - ((5+8)^2-2)

825 = 528 + 52*8 - (2+5)*(25-8)

825 = 285 + 2*85 + 2*5*(52-5*(5-2))

825 = 258 + 25*8 + 2*58 + ((8-5)*85-(8/2))

825 = 8*25 + 82*5 + (8*5+(5-2))*5

825 = 5*28 + 52*8 + 5*2*8 + (2+5)*(5-2)^(8-5)

825 = (8+25) + (82+5) + 5*(5-2)*(2^5+(8+2+5))

825 = (5+28) + (52+8) + (25-8-5)*(58+(8-5))

825 = (8+2+5)*(58-(8-5))

825 = (8*2+5)*8*5 - (8+2+5)

825 = (8+2*5)*5*(8-5)^2 + (8+2+5)

825 = (82+5)*(8-5)^2 + (8*5+2)

825 = (82-5)*(8*2-5) - (25-(5-2))

825 = (8+25)*25

825^3 +/- (825+1) are both prime.

237th Day of the Year.

237 = 273 - 2*(7*3-3)

237 = 327 - 3*27 - (23-2*7)

237 = 372 - 37*2 - (37+(7-3)!)

237 = 732 - 7*32 - (273-2)

237 = 723 - 72*3 - 27*(3+7)

237 = 2*37 + 23*7 + (7-3-2)

237 = 7*32 + 73*2 - 7*(7*3-2)

237 = (2+37) + (23+7) + 2^3*7*3

237 = (7+32) + (73+2) + 3*(37+(7-3))

237 = (2+3+7)*(2*(3+7)) - (2^3-(2+3))

237 = (2*3+7)*(23-(2+3)) + (3^2-3*2)

237 = (2+3*7)*(3+7) + 7*(3-2)

237 = (2+37)*(7-(3-2)) + (7+2)/3

237 = (23+7)*2^3 - (27-23)

237 = (23-7)*(3*(7-2)) - (2^3-(2+3))

237^2 does not contain a 2, 3, or 7.

237^2 +- 2 are consecutive primes.

237^6 + 237^5 + 237^4 + 237^3 + 237^2 + 237^1 + 1 is prime.

August 26th (238th Day of the Non-Leap Year)

Aug 26th, 8/26.

826 = 862 - 6*(8-2)

826 = 682 + 68*2 + 2^(6/2)

826 = 628 + 62*8 - (286+(8*2-(6-2)))

826 = 268 + 26*8 + (8+2)*(6^2-8/(6+2))

826 = 286 + 28*6 + 2*6*(28+6/2)

826 = 8*26 + 82*6 + 6*(28-(8/2)-(6/2))

826 = 6*28 + 62*8 + 2*(6/2)^(8/2)

826 = (8+26) + (82+6) + (8+6/2)*8^2

826 = (6+28) + (62+8) + 2*(8*2+6/2)^2

826 = (8+2+6)*2*26 - (8*2-8-2)

826 = (8*2+6)*(6^2+8/(6+2)) + (6-2)!/2

826 = (8+2*6)*(8^2-(8/2)!) + ((6-2)!+2)

826 = (8+26)*(6-2)! + (8*2-6)

826 = (82+6)*(6*2-6/2) + (8+26)

826 = (82-6)*(8+6/2) - (8*2-6)

238th Day of the Year.

238 = 283 - (3+2)*3^2

238 = 328 - 3*28 - (38-32)

238 = 382 - 38*2 - (8/2)*(2^3+3^2)

238 = 832 - 8*32 - 2*(2+3+8)^2

238 = 823 - 82*3 - (382-(38+(8-3)))

238 = 2*38 + 23*8 - (8*3-2)

238 = 8*32 - 83*2 + (8/2)*(32+3+2)

238 = (2+38) + (23+8) + (23*8+8+3^2)

238 = (8+32) + (83+2) + (8*3^2+(38+3))

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

238 = (2*3+8)*(8+3^2)

238 = (2+3*8)*3^2 + (28-(8/2)!)

238 = (2+38)*2*3 - (8-3*2)

238 = (23+8)*(8-(3-2)) + 3*(8-(3-2))

238 = (23-8)^2 + (2+3+8)

238^2 does not contain a 2, 3, or 8.

238 = 2*7*17 and 238*2717 = 646646 (palindrome concatenating two palindromes).

238 is the sum of the first 2+3+8 = 13 primes.

238^4 + 1 is prime.

238^6 + 238^5 + 238^4 + 238^3 + 238^2 + 238^1 + 1 is prime.

238^16 + 239^16 is prime.

238^23 + 238^21 + 238^19 + ... + 238^5 + 238^3 + 238^1 + 1 is prime.

If 238 = n, (7+n!)/7 is prime (238 is the largest number with this property).

August 27th (239th Day of the Non-Leap Year)

Aug 27th, 8/27.

827 = 872 - (7-2)*(7+2)

827 = 728 + 72*8 - (7+2)*(8*7-8-(7-2))

827 = 782 + 78*2 - (8-(7-2))*(28+(2+7))

827 = 287 + 28*7 + 8*(27+8*2)

827 = 278 + 27*8 + (2+7)*(27+2+8)

827 = 8*27 + 82*7 + ((8-2)^2+(8-7))

827 = 7*28 + 72*8 + (7-2)*(8*2-(7-2))

827 = (8+27) + (82+7) + (728-(7-2)^2)

827 = (7+28) + (72+8) + 8*(2+87)

827 = (8+2+7)*7^2 - (7*2-8)

827 = (8*2+7)*(8-2)^2 - (2-(8-7))

827 = (8+2*7)*(28+2+8) - (8*2-7)

827 = (8+27)*(8/2)! - (8+7-2)

827 = (82+7)*(8*2-7) + (8*2+8+2)

827 = (82-7)*(27-8*2) + 2*(8-7)

827 is prime and is the concatenation of the first two primes cubed.

827^2 + 827 + 1 is prime.

827 divides every possible Fibonacci sequence.

828^11 - 827^11 is prime.

827 concatenated with the next 4 primes (827//829//839//853//857) is prime. (One other prime > 827 has this property.)

827, 827+2, 827+222, and 827+2222 are all prime.

827^2 = 683929 (683 and 929 are also primes).

There are 827 Ulam numbers less than 10^4.

239th Day of the Year.

239 = 293 - 2*9*3

239 = 329 - 3*29 - (9-3*2)

239 = 392 - 39*2 - (9*3-2)*3

239 = 923 - 92*3 - 2^3*(3*(2^3+9))

239 = 932 - 9*32 - (3+2)*3^(9-3-2)

239 = 2*39 + 23*9 - 23*2

239 = 9*32 + 93*2 - (2+3)*(39+2^3)

239 = (2+39) + (23+9) + 2*(92-9)

239 = (9+32) + (93+2) + (2*3*9+(9-2)^2)

239 = (2+3+9)*(2^3+9) + (9-2^3)

239 = (2*3+9)*2^(9-3-2) - (9-3!-2!)

239 = (2+3*9)*2^3 + (9*2-(9+2))

239 = (2+39)*2*3 - (39-32)

239 = (23+9)*2^3 - (2^3+9)

239^3 + 239 + 1 is prime.

239^4 + 239^3 + 239^2 + 239^1 + 1 is prime.

There are 239 primes < 1500.

239^16 + 240^16 is prime.

239^4 - 239 - 1 is prime.

239^4 + 239 + 1 is prime.

239 is related to pi: pi/4 = 4*arctan(1/5) - arctan(1/239).

239 is the smallest prime factor of 1234567654321.

In the Simpson's episode "Homer's Night Out", Homer weighs himself at 239 pounds and starts to exercise.

arctan(1/239) begins 0.239... (only positive integer with this property).

1+3+5+7+9+...+239 = 239+241+243+...+337 (Note both 239 and 337 are prime).

August 28th (240th Day of the Non-Leap Year)

Aug 28th, 8/28.

828 = 882 - (8-2)*(8+8/8)

828 = 288 + 2*88 + (8/2)*(88+(8/2-8/8))

828 = 288 + 28*8 + (8/2)*(82-(8/2-8/8))

828 = 82*(8+2) + (8*2-8)

828 = 82*(8-2) + 8*(28+8+8-2)

828 = 28*(8+8) +(8+2)*(28+2+8)

828 = 82*(8+8) - (28-(8-2))^2

828 = 82*28 - 882 - 2*(288+(8-2-8/8))

828 = 8*28 + 82*8 - 2*(28-2)

828 = (8+28) + (82+8) + (88+2) + (8+28)*(8*2+8/8)

828 = (8+2+8)*(28+(8+8+2))

828 = (8*2+8)*(28+(8-2)) + (8+8/2)

828 = (82-8)*(8+2) + 88*(8/8)

828 = (82+8)*(8+2+8)/2 + (8+2+8)

828 = 2*2*3*3*23 (only uses 2 and 3).

There are 828 primes between 180000 and 190000.

828^19 + 828 + 1 is prime.

The concatenation of 828^3, 828^2, 828^1, 828^0 is prime.

240th Day of the Year.

240 = 204 + (2+4)^2

240 = 420 - 4*20 - (2*4+2)^2

240 = 402 - 40*2 - (40+42)

240 = 24*(2+4) + (2+4)*2^4

240 = 42*(4+2) - 4!/2

240 = 24*2*4 + 4!*2

240 = 42*4*2 - 4!*4

240 = (2+40) + (20+4) + (2+4+0) + 4*42

240 = (2+4+0)*40

240 = 2*4*(20/4)*(2+4)

240 = 2*(20/4)!

240^2 + 1 is prime.

240^8 + 1 is prime.

240^3 + 240 + 1 is prime.

240^4 + 240 + 1 is prime.

There are 240 primes < 39^2.

The 5th derivative of (x^x)^x evaluated at x = 1 is 240.

240 has 20 divisors (more than any number < 240).

August 29th (241st Day of the Non-Leap Year)

Aug 29th, 8/29.

829 = 892 - 9*(8*2-9)

829 = 928 - 9*28 + 9*(28-9-2)

829 = 982 - 98*2 + (29+2*(9-2))

829 = 289 + 28*9 + 2^2*8*9

829 = 298 + 29*8 + (29-(8-2))*(9+8/2)

829 = 8*29 + 82*9 - (9-(8-2))*(29+2*9)

829 = 9*28 + 92*8 - (2+9-8)*(29+(8/2)!)

829 = (8+29) + (82+9) + (8+2+9) + 2*(29+2)*(2+9)

829 = (9+28) + (92+8) + (8/2)*(8*2*9+29)

829 = (8+2+9)*(29+(8*2-2)) + (8*2-8/2)

829 = (8*2+9)*(29+8/2) + (8/2)*(9-8)

829 = (8+2*9)*(28+8/2) - (9-(8-2))

829 = (8+29)*(29-(9-2)) + (9+8-2)

829 = (82+9)*9 + (28-9*2)

829 = (82-9)*(2+9) + (8+2*9)

829*(8+2+9) and 829+(8+2+9) are palindromes.

241st Day of the Year.

241 = 214 + (4-1)^(2+1)

241 = 142 + 1*42 + (4-1)*(21-2)

241 = 124 + 1*24 + (2+1)*(41-2*(4+1))

241 = 421 - 4*21 - (2+4)*2^4

241 = 412 - 41*2 - (4+1)*2^4 - 2^(4-1)

241 = 2*41 + 24*1 + (4+1)*(4!+2+1)

241 = 1*42 + 14*2 + 2^(4-1)*(21-2)

241 = (2+41) + (24+1) + (124+(2*4-1)^2)

241 = (1+42) + (14+2) + 14*(2^4-2-1)

241 = (2+4+1)*(41-2-4-1) + (4-(2-1))

241 = (2*4+1)*(24+1) + 2^4*1

241 = (2+4*1)*41 - (21-2^4)

241 = (2+41)*(2+4*1) - (2^4+1)

241 = (24+1)*(2*(4+1)) - (2*4+1)

241^2 - 241 - 1 is prime.

n! never ends in 241 zeros.

Americium-241 is the most common isotope of Americium. It is used in smoke detectors.

A band named Reel Big Fish wrote a song title "241" where it just repeats the number as the only lyric.

If A = 1, B = 2, C = 3, ... Z = 26, "THE RIEMANN HYPOTHESIS" is prime (241).

August 30th (242nd Day of the Non-Leap Year)

Aug 30th, 8/30.

830 = 803 + 3^3

830 = 308 + 30*8 + 3!*(38+8+3^0)

830 = 380 + 3*80 + 30*(8-3^0)

830 = 83*(8+3) - 83*(3+8-38^0)

830 = 38*(3+8) + (3+8^0)*(8*3+83-(3+8^0))

830 = 38*3*8 - (83-83^0)

830 = 83*(8-3)*(3-8^0)

830 = 38*(8-3) + 8*(83-3)

830 = (8-3)!*3 + (8-3)*(83+8+3)

830^16 + 1 is prime.

3^830, 3^831, 3^832 all have the same number of digits.

830^33 + 830^31 + 830^29 + ... + 830^5 + 830^3 + 830^1 + 1 is prime.

242nd Day of the Year.

242 = 224 + (24-(2+4))

242 = 422 - 4*22 - 4*(4!-2/2)

242 = 422 - 42*2 - (2+4)*2^4

242 = 24*(2+4) + 2*(2*4-2/2)^2

242 = 24*2*4 + (2*4+2)*(4+2/2)

242 = 42*(4-2) + 2*(2*42-4-2/2)

242 = 42*4*2 - 2*(24*2-2/2)

242 = 42*(4+2) - (2*4+2)

242 = 2^(4+2) + 2*(2*42+(4+2/2))

242 = 2^(4*2) - (2^4-2)

242 = 2*42 + 24*2 + (22/2)*(2*4+2)

242 = 22*4 + (2+42) + (24+2) + 2*42

242 = (2+4+2)*(24+2+4) + (2*4-4)

242 = (2*4+2)*24 + 4/2

242 = (2+42)*(4+2/2) + (24-2)

242 = (24+2)*(2*4+2/2) + 2^4/2

242 = (24-2)*(22/2)

242^4 + 1 is prime.

242^8 + 1 is prime.

242, 243, 244, 245 have the same number of divisors (smallest run of 4 numbers).

10^242 + 27 is prime.

There are 242 primes of the form 1 + x^128 for x < 10^5.

There are 242 5-digit perfect powers.

August 31st (243rd Day of the Non-Leap Year)

Aug 31st, 8/31.

831 = 813 + 3*(8-(3-1))

831 = 381 + 3*81 + (8+1)*(8*3-1)

831 = 318 + 31*8 + (8-3)*(38+3*(8-3))

831 = 138 + 13*8 + 31*(13+3!)

831 = 183 + 1*83 + (8-3)*(138-(8*3+1))

831 = 8*31 + 83*1 + (8-3)*(8+3-1)^(8/(3+1))

831 = 1*38 + 13*8 + 13*(18+(8-1)*(8-3))

831 = (8+31) + (83+1) + (8+3+1)*(83-8*3)

831 = (1+38) + (13+8) + 3*(183+(3-1)*(38-1))

831 = (8+3+1)*(83-(8-1)*(3-1)) + (8-(3!-1))

831 = (8+3*1)*(83-8) + (8-(3-1))

831 = (8*3+1)*(38-(8-3)) + (8-(3-1))

831 = (8+31)*(3*(8-1)) + (8+3+1)

831 = (83+1)*(8+3-1) - (3+3!)

831 = (83-1)*(18-8) + (8+3*1)

831 = 3*277 and 831+(3+277) = 1111 (unit repdigit).

831^11 + 831^9 + 831^7 + 831^5 + 831^3 + 831^1 +1 is prime.

831 is a semiprime.

831^2 +- 2 are also both semiprimes.

There are 831 composite numbers <= 10^3.

831 could be used as internet slang for the phrase with "8 letters, 3 words, 1 meaning": I love you.

243rd Day of the Year.

243 = 3^(2*4-3)

243 = 234 + (2+4+3)

243 = 324 - 3*24 - (2+4+3)

243 = 342 - 34*2 - (23+2^3)

243 = 432 - 4*32 - (43+(2+4)*3)

243 = 423 - 42*3 - (2+4)*3^2

243 = 2*43 + 24*3 + (2+3)*(34/2)

243 = 3*42 + 34*2 + (4+3)^2

243 = (2+43) + (24+3) + (23-4)*3^2

243 = (3+42) + (34+2) + (2+4)*3^3

243 = (2+4+3)*(24+3)

243 = (2*4+3)*(24-3) + 24/(4-2)

243 = (2+4*3)*(3^2+2^3) + (2*4-3)

243 = (2+43)*(2+3) + (24-(2+4))

243^5 is the smallest of 243 > 1 that has a 2.

243^3 + 243 + 1 is prime.

243^2+2 and 243^3+2 are prime.

After just 2 moves, a Rubik's Cube can have 243 different orientations.

243 divides 2^243 - 1.

One Venus Day is around 243 Earth Days.

6000...0001 (243 0's) is prime.

September 1st (244th Day of the Non-Leap Year)

Sep 1st, 9/1.

91 = 1+2+3+4+5+6+7+8+9+10+11+12+13

91 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2.

90*91+1 is prime.

91^2 = 8281 (concatenation of consecutive numbers)

91 contains all odd numbers.
91^3 = 753571 has all odd numbers as well.

Of the first 100 primes, 91 do not contain a 0.

1 + 1/2 + 1/3 + ... + 1/91 has a prime numerator.

There are 91 primes between 57000 and 58000.

91 in base 4 is 1123 (Note which type of sequence that is).

91 in base 8 is 133. 91 in base 5 is 331.

6^3 - 5^3 = 4^3 + 3^3 =91.

91 is the largest amount of money possible if you have one coin each less than a dollar (1+5+10+25+50 = 91).

244th Day of the Year.

244 = 424 - 4*24 - 2*42

244 = 424 - 42*4 - (2^4-4)

244 = 24*(2+4) + (2+4+4)^2

244 = 42*(4+2) - (4*4/2)

244 = 42*4*2 - 4*(24-4/4)

244 = 24*2*4 + 4*(2*4+4+4/4)

244 = 2*44 + 24*4 + (44+4*4)

244 = 44*(4+4) - (24/4)*(2+4*4)

244 = 24*2^4 - (2+4+4)*(2^4-2)

244 = 42*2^4 - 4*(2*44+2^4+(2+4/4))

244 = (2+44) + (24+4) + (2+4+4)*(24-(2*4-4/4))

244 = (4+42) + (44+2) + 2*4*(4!-(4+4/4))

244 = (2+4+4)*24 + (24/(2+4))

244 = (2*4+4)*(24-4) + 2^4/4

244 = (2+4*4)*(2*4+4) + (24+4)

244 = (2+44)*(4+4/4) + (4^2-2)

244 = (24+4)*(2^4/2) + (24-4)

244^2 does not contain a 2 or a 4.

244^2 + 245^2 + 246^2 is prime.

There are 244 squares < 39^3.

244^5 - 243^5 is prime.

244*n/(244+n) is never prime.

September 2nd (245th Day of the Non-Leap Year)

Sep 2nd, 9/2.

There are 92 ways to place 8 nonattacking queens on an 8 by 8 board.

90*92+37 and 90*92+73 are both prime.

There are 92 ways to place 2 nonattacking bishops on a 4 by 4 board.

There are 92 primes of the form 1+x^64 for x < 10000.

There are 92 primes between 16^3 and 17^3.

The Guinness Book of world record has a 92-character placename: Taumatawhakatangihangakoauauotamateaurehaeaturipukakapikimaungahoronukupokaiwhenuakitanatahu.

The average American child consumer 92 pounds of sugar a year.

245th Day of the Year.

245 = 254 - (2^4-(2+5))

245 = 425 - 4*25 - 5*(24-2*4)

245 = 452 - 45*2 - (2*4+5)*(4+5)

245 = 524 - 52*4 - (52+(4!-5))

245 = 542 - 5*42 - (5-2)*(24+5)

245 = 2*45 + 24*5 + 5*(5+4-2)

245 = 5*42 + 54*2 - (52+(25-4))

245 = (2+45) + (24+5) + (2*4+5)^2

245 = (5+42) + (54+2) + 2*(24+(45+2))

245 = (2+4+5)*(4!-2) + (5-(4-2))

245 = (2*4+5)*(5*4-2) + (2+4+5)

245 = (2+45)*5 + 5*(4-2)

245 = (24+5)*(24/(5-2)) + (2*4+5)

245 = (24-5)*(2*4+5) - (5-(4-2))

245^4 only has 4 distinct digits.

245 is the smallest number with no zeros such that 245^2 has 2 zeros.

245, 246, 247, 248, 249, 251, 252, 253 all don't have a 0 but, when squared, they contain a 0.

(run of 9 consecutive numbers, including 250)

September 3rd (246th Day of the Non-Leap Year)

Sep 3rd, 9/3.

93^6+93^5+93^4+93^3+93^2+93+1 is prime.

93^16+94^16 is prime.

There are 93 primes between 55000 and 56000.

9*10^93-1 is prime. That is 8999...999 with 93 9's is prime.

The digit '0' and '4' appear 93 times each in the first 1000 digits of pi.

United Airlines Flight 93 was one of the airplanes hijacked on September 11, 2001.

246th Day of the Year.

246 = 264 - (24-6)

246 = 426 - 4*26 - 4*(2^4+6/2)

246 = 462 - 46*2 - (4-2)*62

246 = 642 - 6*42 - (2+4+6)^2

246 = 624 - 62*4 - (4+6)*(26/2)

246 = 2*46 + 24*6 + (2^4-6)

246 = 6*42 + 64*2 + 2*(64+6/2)

246 = (2+46) + (24+6) + (24/6)*42

246 = (6+42) + (64+2) + 2*(64+2)

246 = (2+4+6)*(4!-(6-2)) + 24/(6-2)

246 = (2*4+6)*(24-6) - 4!/(6-2)

246 = (2+46)*(4+6)/2 + (6/2)!

246 = (24+6)*2*4 + (42-6^2)

246 = (24-6)*(26/2) + (2+4+6)

246 = 2*3*41 (uses 1,2,3,4) and 2+3+41 = 46, is a substring of 246.

246^2 + 246 +- 1 are twin primes.

You can make 246 different necklaces with 7 white beads and 7 black beads.

September 4th (247th Day of the Non-Leap Year)

Sep 4th, 9/4.

94^2+1 is prime.

94! - 1 is prime.

94^16+1 is prime.

94^4 + 94^3 + 94^2 + 94 + 1 is prime.

94^2 - 94 - 1 is prime.

The 94th decimal digit of pi begins a string of two 1's.

There are 94 ways to place 6 nonattacking queens on a 6 by 7 board.

The length of an NBA court is 94 feet.

247th Day of the Year.

247 = 274 - (7+2)*(7-4)

247 = 472 - 47*2 - (2*4*7+(74+(7-4-2)))

247 = 427 - 4*27 - 4*(24-(2+4))

247 = 742 - 7*42 - (27-24)*(72-(7-2))

247 = 724 - 72*4 - (2+7)*(27-(2+4))

247 = 2*47 + 24*7 - (2*4+7)

247 = 7*42 + 74*2 - (2*4+7)*(2+4+7)

247 = (2+47) + (24+7) + (2*4*7 + (24+(2+4+7))*(7-4))

247 = (7+42) + (74+2) + 2*(74-(7+4+2))

247 = (2+4+7)*(2^4-7)

247 = (2+4*7)*2*4 + (2^4-(7-4)^2)

247 = (2*4+7)*2^4 + (24/4+(7-4-2))

247 = (2+47)*(7-2) + (74-72)

247 = (24+7)*2*4 - (7-4-2)

247 = (24-7)*2*7 + (7+4-2)

247^2 has 2 zeros.

247^2 does not contain a 2, 4, or 7.

247^6 is the next power of 247 > 1 with a 4.

247 + 2*4*7 = 303

247 - 2*4*7 = 191 (Both palindromes).

247*(2*4*7) = 13832 (almost palindromic too!)

There are 247 numbers < 10^3 with 3 prime factors.

247^23 + 247^21 + 247^19 + ... + 247^5 + 247^3 + 247^1 + 1 is prime.

1 + 1/2 + 1/3 + 1/4 + ... + 1/247 has a prime on the numerator.

248^11 - 247^11 is prime.

247 = 50123 - 49876.

(smallest number expressed as a difference between two integers together containing all digits 0-9)

Many people abbreviate "all the time" as 24/7, meaning 24 hours a day, 7 days a week.

September 5th (248th Day of the Non-Leap Year)

Sep 5th, 9/5.

95/19 = 5/1 = 5 (cancelling out the 9's works here).

95^3+95+1 is prime.

95^6+95^5+95^4+95^3+95^2+95+1 is prime.

There are 95 primes less than 500.

95^2-95-1 is prime.

The most assists in a 7-game NBA playoff series is 95, set by Magic Johnson in 1984.

248th Day of the Year.

248 = 284 - (8-2)^2
248 = 428 - 4*28 - 2*(28+(8-2))
248 = 482 - 48*2 - (2+4)*(24-(8/(4*2)))
248 = 824 - 82*4 - 248
248 = 842 - 8*42 - (8-2)*(42+24/4!)
248 = 2*48 + 24*8 - 4*(2+8)
248 = 8*42 + 84*2 - (4/2)^8
248 = (2+48) + (24+8) + 2*(84-(8/(4*2)))
248 = (8+42) + (84+2) + 4*28
248 = (2+4+8)*(2^4+(4-2)) - 24/(8-2)
248 = (2+4*8)*(24/8+4) + (2^4-(2+4))
248 = (2+48)*(48-42-(8-2)/(4+2)) - (8-4-2)
248 = (2*4+8)^2 - 48/(2+4)
248 = (24+8)*8 - (2^4-8)

248^4 + 1 is prime.
248^16 + 1 is prime (consecutive powers of two, as well).

There are no primes between 2480 and 2489.

1999...999 (248 9's) is prime.

248^5 - 248 - 1 is prime.

2^248, 3^248 and 5^248 all have an even digit sum.

248 is the largest known number n such that (10+n!)/10 is prime.

You can make the powers of 2 up to 256 using each digit at most once. {2, 4, 8, 2^4, 4*8, 2*4*8, 2^4*8, 2^8}

September 6th (249th Day of the Non-Leap Year)

Sep 6th, 9/6.

5! - 4! = 96.

90*96+37 and 90*96+73 are both prime.

96^32 + 1 is prime.

96^2 starts with a 9 and ends with a 6.

96^16 + 97^16 is prime.

249th Day of the Year.

249 = 294 - 9*(9-4)

249 = 429 - 4*29 - 4^(9-4-2)

249 = 492 - 49*2 - 29*(9-4)

249 = 942 - 9*42 - (2+4+9)*(29-2*4)

249 = 924 - 92*4 - 9*2*4 - (9-4)*(49-2)

249 = 2*49 + 24*9 - (9-4)*(9+4)

249 = 9*42 - 94*2 + (9*4*2-9-4)

249 = (2+49) + (24+9) + (24+9)*(9-4)

249 = (9+42) + (94+2) + 2*(2+49)

249 = (2+4+9)*2^4 + (9-4-2)^2

249 = (2*4+9)*(24-9) - (9-4-2)!

249 = (2+4*9)*(2+4) + (2^4+(9-4))

249 = (2+49)*4 + (49-4)

249 = (24+9)*(9-2) + 9*4/2

249^3 ends in 249.

249^4 + 249^3 + 249^2 + 249^1 + 1 is prime.

249*2^249 - 1 is prime.

There are 249 8-digit cubes (8 is a cube itself).

249^5 - 249 - 1 is prime.

September 7th (250th Day of the Non-Leap Year)

Sep 7th, 9/7.

97 is the last prime below 100.

97 and 79 are both prime.

There are 97 primes less than 2^9.

97 is conjectured to be the last prime of the form 3^n+2^n (if there exists a higher one, it is larger than 10^16000000).

97^2+98^2 is prime.

97^2-97-1 is prime.

97, 907, 9007, 90007, and 900007 are all primes.
9000007, 90000007, 900000007, 9000000007, and 90000000007 are all composite.

There are 97 primes with exactly 3 distinct digits.

250th Day of the Year.

250 = 502 - 50*2 - 2*(25*(5-2)+2^0)

250 = 520 - 5*20 - 2*5*(5^2-5-2-52^0)

250 = 205 + 5*(2*5-25^0)

250 = 25*(2+5) + (5-2)*5^2

250 = 52*(5-2) + 2*(52-5)

250 = 52*(5+2) - (25-(5+2^0))*(5+2^0)

250 = 5!*2 + 5*2

250 = (5-2)^5 + (5+2+0)

250 = (5+2)^(5-2) - (5-2)*(25+(5+2^0))

250 = 2^5*(2+5) + 2*(5+2^(5-2))

250 = 2^5*(5-2) + (2*5+25^0)*(25-(2*5+25^0))

250 = (2+5+0)*(25+2*5) + (5+2*0)

250 = 2*5*25 = 2*5*5^2 = 2*5^(5-2)

250^2 + 1 is prime.

250^5 - 250 - 1 is prime.

250^5 + 250^3 + 250^1 + 1 is prime.

There were 250 Pokemon originally available in Gold and Silver, before Celebi.

September 8th (251st Day of the Non-Leap Year)

Sep 8th, 9/8.

10^98+49 is prime.

98^3-98-1 is prime.

98.6 degrees Fahrenheit is normal body temperature.

98 is the highest jersey number offered in the NHL (since 99 is retired by Wayne Gretzky).

251st Day of the Year.

251 = 215 + (5+1)^2

251 = 125 + 12*5 + (5+1)*(2*5+1)

251 = 152 + 1*52 + (51-(5-1))

251 = 512 - 51*2 - (2+1)*(52+1)

251 = 521 - 5*21 - 15*(2*5+1)

251 = 2*51 + 25*1 + (5!+(5-1))

251 = 1*52 + 15*2 + (15-2)^2

251 = (2+51) + (25+1) + (5-1)*(52-(5-2)^2)

251 = (1+52) + (15+2) + (2*5*1) + (2*5-1)*(15+(5-1))

251 = (2+5+1)*(5^2+5+1) + (5-2*1)

251 = (2*5+1)*(25-2) - (5-2-1)

251 = (2+5*1)*(25+2*5) + (5-2*1)!

251 = (2+51)*5 - (5+2)*2

251 = (25+1)*(2*5-1) + (15+2)

251 = (25-1)*(2*5*1) + (2*5+1)

251^3 ends in 251.

251*152 ends in 152.

There are 251 primes less than 40^2.

251^6 + 251^5 + 251^4 + 251^3 + 251^2 + 251^1 + 1 is prime.

251^4 - 251 - 1 is prime. (251 is of the form n^4 - n - 1 too)

251 is prime but 251 +- 2 are both semiprimes.

The 251st Fibonacci number has a digit sum of 251.

The "Vermont 251 Club" is a group of members dedicated to visiting all 251 towns and cities in Vermont, USA.

September 9th (252nd Day of the Non-Leap Year)

Sep 9th, 9/9.

99^2+99+1 is prime.
99^4+99^3+99^2+99+1 is prime.

If 99 divides a four digit number ABCD, then 99 also divides BCDA, CDAB, and DABC.

There are 99 primes less than 23^2.

99^19+99^17+99^15+99^13+99^11+99^9+99^7+99^5+99^3+99+1 is prime.

There are 99 four digit left-truncatable primes.

10^27-99 is the largest prime less than 10^27.

99 = 2^3+3^3+4^3

"99" is a common price ending in psychological pricing.

99 Bottles of Beer is a famous folk song in the United States.

252nd Day of the Year.

252 = 225 + (25+2)

252 = 522 - 5*22 - 5*2*2*(2^(5-2))

252 = 522 - 52*2 - 2*(52+(25+(5-2)!))

252 = 25*(2+5) + (5+2)*(2*5+2/2)

252 = 25*2*5 + (5-2/2)/2

252 = 52*(5+2) - 2^(5-2)*(5*2+2^2)

252 = 52*(5-2) + 2^5*(5-2)

252 = 2*52 + 25*2 + 2*(5+2)^2

252 = (2+52) + (25+2) + (5-2)^2*(25-(5-2)!)

252 = (2+5+2)*(25+(5-2))

252 = (2*5+2)*(25-2^2)

252 = (2+52)*2^2 + (5+2/2)^2

252 = (25+2)*(5-2)^2 + (5-2)^2

252 = (25-2)*(2*5+2/2) - (5-2-2)

252^3 does not contain a 2 or 5.

1 divides 2521, 2 divides 2522, 3 divides 2523, ... 9 divides 2529.

252^4 + 252 + 1 is prime.

1 + 1/2 + 1/3 + 1/4 + ... + 1/252 has a prime on the numerator.

There are 252 ways to place four pieces in a Connect Four board.

September 10th (253rd Day of the Non-Leap Year)

Sep 10th, 9/10.

910 = 901 + 9*1^0

910 = 190 + 1*90 + (9+1)*(9*(9-1-1^0))

910 = 109 + 10*9 + 9*(91-(19-(9-1)+1))

910 = 91*(9+1)

910 = 91*9*1 + (10+9^(1+1))

910 = 19*1*9 + 9*10 + (10+1)*(91-(1+9^0)^(10/(10-(9-1))))

910 = 91*19 - 9*(10+9^(1+1))

910 = (9+10)*(9-1)*(10/(1+1)+1) - (10-(9-1))

910^19 + 910^17 + 910^15 +  ... + 910^5 + 910^3 + 910^1 + 1 is prime.

910+1 and 910^2+1 are both prime.

253rd Day of the Year.

253 = 235 + (23-5)

253 = 352 - 3*(35-2)

253 = 325 - 3*25

253 = 532 - 53*2 - 25*(2+5) - (25-23)

253 = 523 - 5*23 - (25+3!)*5

253 = 2*53 + 25*3 + (2+5)*(5+3)

253 = 3*52 + 35*2 + 3^(5-2)

253 = (2+53) + (25+3) + 2*5*(2+5*3)

253 = (3+52) + (35+2) + 23*(2+5)

253 = (2+5+3)*32 - (53+2*(2+5))

253 = (2*5+3)*(25-3!) + (5+3-2)

253 = (2+5*3)*3*5 - (25-23)

253 = (25-2)*(2*3+5)

253 = (25+3)*(3!+5-2) + (3*2-5)

253 = (2+53)*5 - (25-3)

253 = 1+2+3+...+22.

253 remains in the natural number sieve of the positive integers. (www.oeis.org/A000960)

253 = 11*23 (first four Fibonacci numbers in order).

253^2 does not contain a 2, 5, or 3.

253^2 + 4 is prime.

253^2 = 64009 (concatenation of two squares).

The 253rd and 254th primes are twin primes.

You need 253 people in a room for > 50% chance that someone has the same birthday as you.

25301, 25303, 25307, and 25309 are all prime.

There are 253 squares < 40^3.

Sep 11th, 9/11.

September 12th (255th Day of the Non-Leap Year)

Sep 12th, 9/12.

91*12 = 1092 (only has digit 9,1,2--excluding zero)

912 = 91*12 - (9+1)*(9*2)

912 = 129 + 192 + 291 + 219 + 9^(1*2)

912 = 345 + 567

912 is a multiple of 12 and has a 12 in its decimal representation. (What is the next number with this property?)

912 = 79^2 - 73^2 (73,79 are consecutive primes). (What are the next two consecutive primes that satisfy this equation?)

912 = 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 (10 consecutive primes).

912 = 223 + 227 + 229 + 233 (4 consecutive primes).

In the "Stonecutters" episode of The Simpsons, Lenny told Homer the real emergency number is 912.

255th Day of the Year.

255 = 3*5*17

3 + 5 + 17 = 25 (contained in the decimal representation of 225). (What other numbers satisfy this property?)

255^2 contains 2, 5, and 5.

255^4 + 255^3 + 255^2 + 255 + 1 is prime.

There are 255 levels in PacMan. (Why?)

255 is a repeating digit in base 2 (11111111), base 4 (3333), and base 16 (FF). (What other numbers are repeating digits in three different bases besides base 10? Four different bases?)

September 13th (256th Day of the Non-Leap Year)

Sep 13th, 9/13.

913 = 91*13 - (91*3) - (9*1)/3

91*3 + 9*13 + 1 = 391 (rotation of 913)

913 = 91 + 13 + 93 + 19 + 31 + 39 + 193 + 139 + 319 - (9-1)*3

913 - (9+1+3) = ((9+1)*3)^(3-1)

913 = 11*83

(1+1) + (8+3) = 9+1+3

Also, 319 = 29*11

(2+9) + (1+1) = 3+1+9 (What other numbers have both these properties?)

913+7 = 920

913920 = 960*952 (960 - 952 = 8)

9013 and 9103 are both primes.

913^32 + 914^32 is prime.

913^2 and 914^2 have the same digits. (What is the next set of numbers with this property?)

913 is the sum of the first 43 integers minus perfect fifth powers (1 and 32).

After escaping the Jedi purge, Obi-Wan Kenobi transmitted a 913 emergency transmission.

256th Day of the Year.

256 = 2^(2+6)

256 = 25 + 56 + 26 + 52 + 62 + 65 - 5*6

256 = 2*56 + 25*6 - 6

256 = (6-2)*(2^6)

256 - 2*5*6 = (2*5+ (6-2))^2

256 = (2*5 + 6)^2

256, 256^2, 256^3, and 256^4 do not contain an 8 or a 0.

256^n (n is at least 2) contains at least 2 pairs of the same digit (i.e. 256^2 = 65536--two 5s and two 6s)

256 = 2^8 = 16^2 = 4^4 = (4^2)*(2^4) = ((2^2)^2)^2 (Note: 2*2 = 4, 4*2 = 8, 8*2 = 16...all are used in these equations).

256^2 + 1 is prime.

There are no primes between 2560 and 2569.

256 = 16^2

25 = 5^2 (removed last digit. Note: |1-6| = 5)

256 has 9 divisors. (Exercise. Prove that if a number has 9 divisors, it is a perfect square.)

There are 256 NFL regular season football games.

September 14th (257th Day of the Non-Leap Year)

Sep 14th, 9/14.

914 = 9*14 + 91*4 + 149 + 194 + 9^((9-1)/4)

914 = 419 + 491 + 4

914 = 149 + 194 + 419 + 4*(9*1*4) + (9-1)

914 - 419 = 99 + 88 + 77 + 66 + 55 + 44 + 33 + 22 + 11

914 = 91*14 - 9*1*4*(9+1)

All the digits of 914 are squares. (What's the next composite number with the same property?)

914 = 2*457 (2 ≤ 4 ≤ 5 ≤ 7) (What is the next number to satisfy this?)

There are 914 ethylene derivatives with 11 carbon atoms.

914 = 2^10 - 10^2 - 10

914 = 1*2*457 (divisors)

1 + 2 + 457 = 419 + 41

257th Day of the Year.

257 is prime, is made up of prime digits and the difference between any two digits is also prime. (Are there other numbers like this?)

25*7 + 2*57 = (2*5 + 7)^2
257 = 25 + 57 + 27 + 75 + 2*5*7 + (2*5-7)

752 - 257 = 99 + 88 + 77 + 66 + 55 + 44 + 33 + 22 + 11

257 + (2*5 - 7) = 98 + 76 + 54 + 32

257 = 25*7 + 2*57 - 2^5

257^2 does not contain 2, 5, or 7. (What's the next number with this property?)

257*752 = 193264 (all different digits) (Are there other numbers with the same property?)

257 is a Fermat prime: 257 = 2^(2^3) + 1

257 is a prime satisfying the following equations:

n^n + 1 (n = 4)

16^n + 1 (n = 2)

n^2 + 1 (n = 16)

4^n + 1 (n = 4)

n^4 + 1 (n = 4)

2^n + 1 (n = 8)

n^8 + 1 (n = 2)

n^8 + (n+1)^8 (n =1) (What other primes are of one or more of these forms? Is there another prime satisfying all equations?)

263 is the next prime after 257.

257263 is also prime. (What other two consecutive primes satisfy this property?)

September 15th (258th Day of the Non-Leap Year)

Sep 15th, 9/15.

915 = 3*5*61

3 + 5 + (6+1) = 9 + 1 + 5 = 15 (contained in 915)

915 is divisible by 15. (What's the next number with this property?)

915 = 91*5 + 9*15 + 95 + 15^((9+1)/5) + 5

915 = 591 + 159 + (9 -1+5)^((9+1)/5) + (1-5)

915 + (9-1+5) = 519 + 195 + 9*1*5 + (9-1+5)^((9+1)/5)

915 - 519 = 88 + 77 + 66 + 55 + 44 + 33 + 22 + 11

915*45 starts with a 4 and ends with a 5. (Note: 45 = 9*1*5) (Are there other numbers with this property?)

There are 915 primes less than 10,000 that do not contain an 8.

There are 915 bicentered hydrocarbons with 14 atoms.

915 is a number n such that the concatenation 2n3n5n7n11n13 is prime. Hence, 29153915591579151191513 is prime. (What's the next number with this property?)

916^11 - 915^11 is prime.

2^915, 3^915, 5^915, 7^915, and 11^915 all have even digit sums.

9+1+5 = 15

9*1*5 = 45 (multiple of 15)

915^2 = 837225 (no digits are squares themselves).

258th Day of the Year.

2,5,8 appear on the same column on most phones. It is the last day of the year to do this (first was 147).

258 = 25 + 58 + 28 + 52 + 85 + (2+8)

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

258 = (25+8) + (2+58) + (8+5)^2 + 8/2

258 - 2*8 = 66 + 55 + 44

258^2 contains three of the same number. (What's another number that has this property?)

258*(2+5+8)*(2*5*8) = 309600 (all digits are multiples of 3--excl 0) (Do other numbers have this property?)

258^4 does not contain 2,5, or 8. (What other numbers have this property?)

258 = 59 + 61 + 67 + 71 (4 cons primes)

258 = 211 + 47 (47th prime + 47)

258^10 + 258^9 + 258^8 + 258^7 + 258^6 + 258^5 + 258^4 + 258^3 + 258^2 + 258 + 1 is prime. (What's the next number that satisfies this property?)

258 = 6^3 + 6^2 + 6^1

258 = 1^8 + 1^8 + 2^8 (first 3 Fibonacci numbers to the 8th power)

3^258 + 14 is prime (Note: 3,14 translates to 3.14--pi) (What is the next number with this property?)

September 16th (259th Day of the Non-Leap Year)

Sep 16th, 9/16.

In other countries, today is 16/9. 169 = 13^2 and is concatenated of two squares (4^2 and 3^2). (Are there other numbers like this?)

916 = 619 + 169 + (9-1-6)^(1+6)

916 = 691 + (6+9)^(9-1-6)

916 = 196 + 169 + 619 - (69 - 1)

916 = 91 + 16 + 96 + 61 + 19 + 69 + 91*6 + 9*(9-1-6)

916 = (9+1+6)*(9*1*6) + (61 - 9)

916's digits create three squares.

169 = 13^2

196 = 14^2

961 = 31^2

(What other numbers have this property?)

916*(9*1*6) = 49464 (alternates the digit 4). (What other number has this property?)

916*619 = 567004 (only 4,5,6,7--excl 0's) (What's another number that satisfies this property?)

916^(9+6+1) + 1 is prime. (What's the next number with this property?)

3^916, 3^917, and 3^918 all have the same number of digits. (What's the next number with the same property?)

There are 916 pairs of prime numbers below 10^4 that differ by 10.

There are only 5 primes between 91600 and 91699. (What are they?)

916 in Roman numerals is CMXVI (all different letters).

259th Day of the Year.

259 = 25 + 59 + 29 + 95 + 52 - (2*5) + 9

259 = 25*9 + 2*59 - 2*5*9 + (2 - 5 + 9)

259 = (25+9) + (2+59) + 9^(9-5-2) + 59 + (29-5)

259 - 2*5*9 = 169 (Date for Today and 13^2)

259 = 6^3 + 6^2 + 6^1 + 6^0

259!/9 + 1 is prime.

259 is a divisor of 999999999999.

259 is a repdigit in base 6 (1111).

September 17th (260th Day of the Non-Leap Year)

Sep 17th, 9/17.

917 = 91 + 17 + 97 + 719 - 7

917 = 71 + 19 + 79 + 17 + 91 + 97 + 179 + 197 + 17*9 + (7*1)*(9-7)

917 = 9*17 + 91*7 + (9-7)*(9*1*7) + 1

917 = 777 + (9+1)*(9-7)*7

917*(9*1*7) = 57771 (repeating three 7s in a row). (What other numbers have this property of repeating digits?)

917 = 7*131

9^2 + 1^2 + 7^2 = 131.

Hence, 917 = 7*(9^2 + 1^2 + 7^2) (Only 4 numbers have such property). (What are the other 3 numbers?)

Adding 987654321 to the front of 917 makes a prime. (Hence, 987654321917 is prime). (What is the next number with this property?)

917 = 173 + 179 + 181 + 191 + 193 (5 cons primes)

260th Day of the Year.

260 = 206 + 62 - (6+2)

260 - 26 - (6/2) = 66 + 55 + 44 + 33 + 22 + 11

260 = (26+0) + (2*60) + 62 + 2*26

260^2 + 1 is prime.

10^260 + 49 is prime.

260^4 + 260^3 + 260^2 + 260^1 + 260 is prime.

1 + 260 + 260^3 + 260^5 + 260^7 + 260^9 + 260^11 + 260^13 + 260^15 + 260^17 + 260^19 + 260^21 + 260^23 is prime.

n! never ends in 260 zeroes. (What is the next number with this property?)

There are 260 ways to place 4 non-attacking bishops on a 4 by 4 board.

260 is the magic constant for Benjamin Franklin's 8 by 8 magic square.

September 18th (261st Day of the Non-Leap Year)

Sep 18th, 9/18.

918 = 91 + 18 + 98 + 189 + 198 + 81 + 19 + 89 + 9*1*8 + 9*(8-1)

918 = 91*8 + 9*18 + 9*1*8 - (9+18) - ((9*1)+8)

918 = 2*3*3*3*17

918*(2+3+3+3+17) = 25704 (no 9,1, or 8). (What other numbers have this property?)

918^2, 918^4, and 918^6 all have 3-digit palindromes in them.

918*819 = 751842 (all digits except multiples of 3) (Are there other numbers with this property? All digits except multiples of 4? 5? etc.)

Also 18/9 in other countries.

918*189 = 173502 (all non-composite digits). (Are there other days of the year with this property?)

918^8 + 1 is prime.

918^9 + 918^7 + 918^5 + 918^3 + 918 + 1 is prime.

918^13 + 918^11 + 918^9 + 918^7 + 918^5 + 918^3 + 918 + 1 is prime.

There are 918 four digit numbers whose adjacent digits differ by 2 or less.

918^3 does not have a 9,1, or 8. (What's the next number that satisfies this property?)

All divisors of 918 use every digit 0-9 in their decimal representation. (What other numbers have this property?)

919*920*921*922 / 918+919+920+921+922 is an integer. (What is another number with the same property?)

9*1*8 = 72.

72-1 and 72+1 are prime.

9+1+8 = 19.

19-1 and 19+1 are prime. (What is the next number with this property?)

There are 918 primes between 45^pi and 46^pi.

261st Day of the Year.

261 = 26 + 61 + 21 + 126 + (26+1)

261 = 6^(2+1) + (26+1) + 6*(2+1)

261 = 22 + 66 + 11 + 162

261 = 2*61 + 26*1 + 126 - (2*6)+1

261^2, 261^3, 261^4, 261^5, 261^6, and 261^7 all have three digit palindromes in their decimal representation. (What other numbers have this property?)

261*(2+6+1) = 2349 (2 < 3 < 4 < 9). (Are there other numbers with this property?)

2^261 - 261 is prime.

There are more primes between 261 and 2*261 than 261^2 and 262^2. (What's the next number with this property?)

261! contains 261 and 162 in its decimal representation. (What's the next number with the same property?)

There are 261 primes between 41^2 and 42^2.

September 19th (262nd Day of the Non-Leap Year)

Sep 19th, 9/19.

919 = 199 + 9*(9*9-1)

919 = 91*9 + 9*19 - 9*1*9 + (9+1)

919 = 91*(9+1) + (9/1)

919 = 19*1*9 + (19-(9-1))*(19-9/9-1)*(9-1)/(9/9+1)

919 = 91*9*1 + (91+9)

919 = 991 - 9*(9-1)

919 = 99*19 - 91*9 - (9+9/9+1)*(91/(9-1-9/9))

919 = 91*9 + 9*19 - 9*(9-(9+1)/9)

919 = (9+19) + (91+9) + (99+1) + (9+1+9) + (9+19)*(9-(9+1)/(1+9/9))!

919 = (9+1+9)*(9-1)*(1+9/9+1)! + (9-1-9/9)

919 = (91+9)*91 + (19-(1+9))

919 = (91-9)*(9+1+9/9) + (19-1-9/9)

919 = (9+19)*(1+9/9)^((1+9)/(1+9/9)) + (9+9+(9+1)/(1+9/9))

9+1+9 = 19 (in decimal representation of 919 and also prime). (What's the next number with this property?)

919*(9+19) = 25732 (all prime digits). (What other numbers have this property?)

919, 991, and 199 (all permutations of 919) are all prime. (What's the next number, besides 991, with this property?)

919 is prime.

9^2 + 1^2 + 9^2 is also prime.

9^3 + 1^3 + 9^3 is also prime. (Do other numbers have this property?)

919 is the 100th palindrome.

(3^919 + 16^919) / 19 is prime. (Note: 3+16 = 19 = 9+1+9)

919 = 4^3 + 7^3 + 8^3 (Note: 4+7+8 = 9+1+9) (Are there other numbers with this property?)

919 is prime and its digits are squares. (What's the next number with the same property?)

919 = 1^3 + 4^3 + 5^3 + 9^3 and 1459 = 9^3 + 1^3 + 9^3.

262nd Day of the Year.

262 = 622 - 62*2 - (6-2)*(62-6/2)

262 = 226 + 2*(26-(2+6))

262 = 26 + 62 + 22 + 26*2 + (2+6+2)^2

262 = 26*2 + 2*62 + 62 + (26-2)

262 = 22*6 + (26+2) + (2+62) + (2+2)*6 + (2*6+2)

262 = 26*(2+6) + (6-2)*(6+6/2)

262 = 26*2*6 - (62-6*2)

262 = 62*(6+2) - (26/2)*(26-2-6)

262 = 62*(6-2) + (2*6+2)

262 = 62*6*2 - 2*6*2 - 2*(226+6/2)

262 = 2*6^2 + (2+6+2)*(6^2/2+2/2)

262 = 2*2^6 + 2*(62+(6-2/2))

262 = 2^(6+2) - 6*2/2

262 = (26+2) + (2+62) + (2+6+2)*(26-(2+6-2/2))

262 = (2+6+2)*26 + (6-2-2)

262 = (2*6+2)*(26-2-6) + (2+6+2)

262 = (2+62)*(6-2) - (6/2)!

262 = (26+2)*(6*2-6/2) + (2*6-2)

262 = (26-2)*(2*6-2/2) - (6-2-2)!

262 is a palindrome and 262^2 has a 3-dig palindrome in its decimal representation.

262^2 + 263^2 + 264^2 is prime.

Today is the 262nd day of the year on 9/19 (both numbers are palindromes).

September 20th (263rd Day of the Non-Leap Year)

Sep 20, 9/20.

920 = 290 + 209 + 9*20 + 2*90 + 92 - 29 - 2

920 = 92*(9+2) - 29 - 9*(9-2-0)

920 - 9*2 = 902 (same digits as 920). (What's the next number with this property?)

920^2 + 1 is a twin prime with 920^2 + 3. (What's the next number that satisfies this property?)

920^2 has only even digits.

There are 920 distinct palindromes in the form of an 11-digit number times its reverse.

There are 920 ways to place 3 checkers on a 5 by 5 board.

263rd Day of the Year.

263 = 26 + 63 + 23 + 36 + 62 + 32 + 2*6 + (6+3)

263 = 26*3 + 2*63 + (62-3)

263 = 236 + 26 + (6-3-2)

263^2 = 69169 (starts/ends with same 2-dig number). (Are there other numbers with this property?)

There are 263 primes less than 41^2 (Note: 263 and 41 are both prime).

2^263 + 7 is prime.

The prime before 263 is 257. 257263 is prime (concatenation of 257 and 263). (What other consecutive primes have this property?)

There are more primes between [263, 2*263] than [263^2, 264^2]. (What other numbers satisfy this property?)

263 is an irregular prime, an Eisenstein prime, a long prime, a Chen prime, a Gaussian prime, a happy prime, a sexy prime, a safe prime, and a Higgs prime.

263 = 11 + 101 + 151 (three primes that all start/end with 1)

263!! has 263 digits.

Anne Frank and her family hid from the Nazis in an annex behind the house at Prinsengracht 263 in Amsterdam.

263 is the largest known prime whose square is strobogrammatic (same number when read upside-down).

September 21st (264th Day of the Non-Leap Year)

Sep 21, 9/21.

921 = 129 + 219 + 291 +192 + 9*(9+2-1)

921 = 92 + 21 + 91 + 29 + 12 + 19 + 291 + 219 + 129 + 9*2*1

921 = 9*21 + 92*1 + 91*2 + 219 + 192 + 19*2 + 9

921*9*2*1 = 16578 (contains 4 consecutive numbers) (Are there other numbers like this?)

921, 922, and 923 are all the product of two primes.

921, 922, and 923 have the same number of divisors.

1 + 921 + 921^3 + 921^5 + 921^7 + 921^9 + 921^11 + 921^13 + 921^15 + 921^17 + 921^19 + 921^21 + 921^23 + 921^25 + 921^27 + 921^29 + 921^31 + 921^33 + 921^35 + 921^37 + 921^39 is prime. (Next number like this?)

The sum of the first 921 odd primes is palindromic (3091903). (What's the next number of primes?)

921 = Floor[e^14/14^e]

There are 921 10-digit numbers divisible by 5^10.

9*2*1 = 18. (17 and 19 are both prime)

9+2+1 = 12. (11 and 13 are both prime) (What other numbers satisfy this property?)

921^0 + 922^1 + 923^2 + 924^3 + 925^4 + 926^5 + 927^6 + 928^7 + 929^8 + 930^9 + 931^10 + 932^11 + 933^12 is prime. (What's the next number that satisfies this property?)

264th Day of the Year.

264 = 26 + 64 + 24 + 46 + 62 + 42. (Any other numbers with the same property?)

264 = 26*4 + 2*64 + (2+6)*4

264 = 2*6*4 + 6^((2*6)/4)

264 = 99 + 88 + 77 = 33 + 55 + 77 + 99 (3,5,7,9--cons. odds)

264^2 = 69696 (palindrome with only 2 distinct numbers) (Other numbers like this?)

1 + 264 + 264^2 + 264^3 + 264^4 + 264^5 + 264^6 is prime.

264^2 + 1 is prime.

264 is divisible by 2,6, and 4. (Other numbers like this?)

264^16 + 265^16 is prime (Note: 264 has 16 divisors)

24*264/(24+264) is an integer (2nd to last number with this property--10 numbers in all) (What is the last number?)

1 + 264 + 264^3 + 264^5 + 264^7 + 264^9 + 264^11 + 264^13 + 264^15 + 264^17 + 264^19 + 264^21 + 264^23 is prime. (Next number with property?)

264 = 11+13+17+19+23+29+31+37+41+43 (10 cons primes)

1693 (264th prime) - 264 = 1429 (a prime) (What other numbers have this property?)

September 22nd (265th Day of the Non-Leap Year)

Sep 22, 9/22.

922 = 92 + 22 + 229 + 292 + 92*2 + 9*2*2 + (9+22) - (9-2)

922 = 292 + 9*22 +92*2 + 229 + (29-2) - (9 - 2/2)

922^2 and 922^3 do not contain a 9 or 2 (digits of 922).

1 + 922 + 922^3 + 922^5 + 922^7 + 922^9 + 922^11 + 922^13 + 922^15 + 922^17 + 922^19 + 922^21 + 922^23 + 922^25 + 922^27 + 922^29 + 922^31 + 922^33 + 922^35 + 922^37 + 922^39 + 922^41 + 922^43 + 922^45 + 922^47 is prime.

265th Day of the Year.

265 = 26 + 65 + 25 +2*65 + (25-6)

265 = 2*65 + 26*5 + 5

265 = (2+6+5)*(2*6*5) - 526 + (6+5)

265!! has 265 digits.

26*5 = 2*65

265*562 = 148930 (all different digits, none 2,6, or 5)

The sum of divisors of 265 (including 265) is a square (324 = 18^2).

September 23rd (266th Day of the Non-Leap Year)

Sep 23, 9/23.

923 = 92 + 23 + 93 + 392 + 329 - 2*3

923 = 92*3 + 93*2 + 9*23 + 239 + (9+2*3)

923 = 239 + 392 + 293 - 1

923 = 329 + 293 + 239 + (9*2*3) + 9-(3-2)

923*923 and 923*329 both have 3-dig palindromes in their decimal representation. (Do other numbers share this property?)

(9+2)^3 = 1331.

1331923 and 9231331 are prime (concatenations 1331//n and n//1331). (Are there other numbers with the same property?)

The number 9876543210923 is prime (9876543210//923) (What other numbers satisfy this property?)

266th Day of the Year.

266 = 26 + 66 + 62 + 66*2 - (26-6)

266 = 26*6 + 2*66 - 26 + (6-2)

There are 266 primes less than 1700.

1 + 266 + 266^2 + 266^3 + 266^4 + 266^5 + 266^6 is prime. (What is the next number with this property?)

1*(266)^0 + 2*(266)^1 + 3*(266)^2 is prime. (What other numbers have this property?)

In base 11, 266 is 111 (repdigit). (What's the next number that satisfies this property?)

September 24th (267th Day of the Non-Leap Year)

Sep 24, 9/24.

924 = 92 + 24 + 94 + 429 + 249 + 9*4

924 = 429 + 492 + (9-2-4)

924 = 492 + 294 + 249 - 9*2*4 - (42-(9-2-4))

924 = 92*4 + 9*24 + 94*2 + 9*2*4 + 92 - 4*(9-2-4)

924^2 = 853776 (not 9,2, or 4). (What is the next number with this property?)

924^3 contains a run of four 8s. (Do other numbers share this property?)

924*429 = 396396 (same 3-dig number). (Are there other numbers with this property?)

924 = 2^10 - 10^2

924 = 2*2*3*7*11

924*(2+2+3+7+11) = 23100 (924 is divisible by 231). (What other numbers have this property?)

There are 924 primes between 60000 and 70000.

There are 924 5-digit primes beginning with 5.

924 = LCM(44,42)

2^924, 3^924, 5^924, 7^924, 11^924 all have even digit sums. (What's the next number with this property?)

924^32 + 925^32 is prime. (What other numbers share the same property?)

267th Day of the Year.

267 = 26 + 67 + 27 + 2*6*7 + 62

267 = 62 + 76 + 72 + (62-7) + 2

267 - 6 = 98 + 76 + 54 + 32 + 1

267 - 6*7 = 1 + 23 + 45 + 67 + 89

267^2 ends in 289 (17^2). (What other numbers, when squared, end in a square?)

1 + 267 + 267^3 + 267^5 + 267^7 + 267^9 + 267^11 + 267^13 + 267^15 + 267^17 + 267^19 is prime. (Other numbers with this property?)

There are 267 groups of order 2^6.

There are 267 primes between 30^3 and 31^3.

2^2 + 6^2 + 7^2 = 89.

267 is divisible by 89. (What is another number with this property?)

September 25th (268th Day of the Non-Leap Year)

Sep 25, 9/25.

925 = 92 + 25 + 95 + 529 + 9*2*5 + 92 + (9-2-5)

925 = 592 +259 + 52 + (29-5) - (9-2-5)

925 = 529 + 295 + 95 + (9+2-5)

925 = 92*5 + 9*25 + 95*2 + (52-9) + (9-2)

925*529 = 489325 (9,2,5 are in order). (Do other numbers have this property?)

925^3 - 925^2 - 1 and 925^3 - 925^2 + 1 are twin primes. (What other numbers are like this?)

If 925 = x, x^4 + 4x^3 + 12x^2 + 24x + 24 is prime (note: (x^4)' = 4x^3, (4x^3)' = 12x^2, etc..) (What's the next number satisfying this property?)

268th Day of the Year.

268 = 26 + 68 + 28 + 2*6*8 + 62 + (2-6-8)

268 = 86 + 62 + 82 + (28-6) + (2+6+8)

268 = 26*8 + 2*68 - (82-6)

268^10 + 268^9 + 268^8 + ... + 268^2 + 268 + 1 is prime.

268^19 + 268^17 + 268^15 + ... + 268^5 + 268^3 + 268 + 1 is prime.

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

268th prime - 268 is prime. (1721 - 268 = 1453).

268 is a multiple of 4.

2+6+8 = 16 is also a multiple of 4. (What's the next number with this property?)

268 is base 8 is 414 (palindrome). (What's another number with this property?)

September 26th (269th Day of the Non-Leap Year)

Sep 26, 9/26.

Today is also 26/9 in other countries.

On the 269th day of the year, it is 26/9. No other date is like this.

926 = 962 - 9*(6-2)

926 = 692 + 69*2 + 6*(2^(6-2))

926 = 629 + 6*29 + (6/2)*(29+2*6)

926 = 296 + 2*96 + 6*(92-9*2-(9-6-2))

926 = 269 + 26*9 + 9*(29+2*9)

926 = 92 + 26 + 96 + 629 + (92-6) + (9-6)

926 = 62 + 29 + 69 + 692 + (62+9) + (9-6)

926 = 9*26 + 92*6 + (9-2)*(9*2+2)

926 = 6*29 + 62*9 + 2*(96+(9-6-2))

926 = 296 + 629 + (9-2-6)

926 = 269 + 692 - (9+26)

926 = 92*6 + 9*26 + 9*2*6 + (29+6) - (9-6)

926 = (9+26) + (92+6) + (69-2^(9-6))*(9+6-2)

926 = (6+29) + (62+9) + (6*2+6+2)*(29+2*6)

926 = (9+2+6)*9*6 + 2^(9-6)

926 = (9*2+6)*(26+2*6) + 2*(9-2)

926 = (9+2*6)*(29+9+6) + (26-(6-2)!)

926 = (9+26)*26 + 2*2^(9-6)

926 = (92+6)*9 + (6-2)*(9+2)

926 = (92-6)*(9+2) - (26-(6/2)!)

926^2 contains 4,5,6,7,8 (a run of numbers). (Do other numbers have this property, as well?)

926*(9*2*6) = 100008 (a run of four 0s). (Are there other numbers with this property?)

926 - (9+2+6) = 909.

926 - (9*2*6) = 818. (both palindromes) (What other numbers share this property?)

926^32 + 927^32 is prime. (What's the next number with this property?)

There are 926 ways to place 7 nonattacking kings on a 7 by 4 board.

926 = 15*16 + 17*18 + 19*20

926 = 139 + 149 + 151 + 157 + 163 + 167 (6 cons primes)

269th Day of the Year.

269 = 296 - (6/2)^(9-6)

269 = 629 - 6*29 - (9-6)*62

269 = 692 - 69*2 - (9+6)*(26-(9-2))

269 = 26 + 69 + 29 + 2*6*9 + (26+9) + 2

269 = 96 + 62 + 92 + (2+6+9) + 2

269 = 962 - 9*62 - 9*(9+6)

269 = 926 - 92*6 - (9+6)*(9-2)

269 = 26*9 + 2*69 - (2+96) - (2+9-6)

269 = 9*62 - 96*2 - (96+(9-6-2))

269 = (2+69) + (26+9) + (2+6+9) + 2*(96-(29-6))

269 = (9+62) + (96+2) + (9*2-6-2)^2

269 = (2+6+9)*(6+9) + 2*(9-2)

269 = (2+6*9)*(6-2) + 9*(9-(6-2))

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

269 = (26+9)*(9-2) + (6-2)!

269 = (2*6+9)*(9+6-2) - (6*2-(9-2))

269^4 contains the numbers 1,2,3,4,5, and 6 (a total of 10 digits). (Do other numbers have this same or similar property?)

269 - (2+6+9) = 252.

269 - (2*6*9) = 181. (both palindromes) (What's another number with this property?)

269 is prime.

269^16 + 270^16 is prime.

2069 and 2609 are also prime.

269^23 + 269^21 + 269^19 + ... + 269^7 + 269^5 + 269^3 + 269 + 1 is prime.

(What other numbers satisfy one or more of these properties?)

September 27th (270th Day of the Non-Leap Year)

Sep 27, 9/27.

927 = 972 - 9*(7-2)

927 = 792 + 79*2 - (9*2+(7-2))

927 = 729 + 7*29 - (97-92)

927 = 279 + 27*9 + (7-2)*(27/9)^(9+2-7)

927 = 297 + 2*97 + 2*(279-(72-(9+2)))

927 = 29 + 72 + 79 + 729 + (9+2+7)

927 = 92 + 27 + 97 + 297 + 279 + (9*2*7) + 9

927 = 9*27 + 92*7 + (29+2+9)

927 = 7*29 + 72*9 + (79-(27/9))

927 = 792 + (97*2) - (72-9) + (9+2-7)

927 = (9+2-7) + 92*7 + 9*27 + (27+9)

927 = (9+27) + (92+7)*9

927 = (7+29) + (72+9) + 27*(29+9/(7+2))

927 = (9+2+7)*(29*2-7) + (7*2-(7-2))

927 = (9+2*7)*(29+2+9) + (9*2-9-2)

927 = (9*2+7)*(27+2*(7-2)) + (2*(9/(7+2)))

927 = (9+27)*(27-2) + (29-2)

927 = (92+7)*9 + (9+27)

927 = (92-7)*(9+2) - (9-9/(2+7))

927 - 792 = 135 and

927 - 279 - 297 = 351. (What other numbers share this property?)

927^19 + 927^17 + 927^15 + ... + 927^5 + 927^3 + 927 + 1 is prime.

927^35 + 927^33 + 927^31 + ... + 927^5 + 927^3 + 927 + 1 is prime. (Are there other numbers with one or both of these properties?)

10^927 - 7247 is prime (7247 is the 927th prime).

There are 927 ways to place 4 nonattacking bishops on a 4 by 5 board.

270th Day of the Year.

270 = 207 + 20*7 - 7*(7+2^2)

270 = 720 - 7*20 - (7+2+72^0)*(70/2-(7-2-2))

270 = 27 + 70 + 20 + 2*72 + (2+7+0)

270 = 27*(2+7) + (7-2-2)^(27/(2+7))

270 = 27*2*7 - (70+7+2+2)

270 = 72*(7-2) - (2+7)*(2+7+27^0)

270 = 2*70 + (20-7)*(2*(7-2))

270 = 207 + 72 - (2+7+0)

270 = (2+7+0)*(27+(7-2-2))

270 = (2+70)*(7-2-2) - (7+2)*(7-2^0)

270 = (20-7)*20 + (2+7+27^0)

270 = (70-2)*(7-2-2) - (20-2*(2+7))

270 = 98 + 76 + 54 + 32 + 10

270^2 = 72900 (today's date is 927, the reversal of 72900). (For what other dates, if any, does this work?)

270 = 2*3*3*3*5

270*(2+3+3+3+5) = 4320 (4  > 3  > 2  > 0). (Do other numbers have this property?)

270^1 + 1 is prime.

270^2 + 1 is prime. (What other numbers satisfy this property?)

The sum of the digits in 58! is 270.

270^2 - 1733 is prime (1733 is the 270th prime).

There are 270 primes in the interval [43, 43^2].

10! has 270 divisiors.

270 is the number of U.S. Electoral College votes needed to be President of the United States.

270 is the smallest number such that its one of its divisors ends with a 0,1,...or 9. (What's the next number like this?)

September 28th (271st Day of the Non-Leap Year)

Sep 28, 9/28.

928 = 92 + 28 + 98 + 289 + 298 + 9*2*8 - (29-8)

928 = 82 + 29 + 89 + 829 - 9*2*8 + 2*(9+8) + 9

928 = 9*28 + 92*8 - (9+28) - (9+(2*8)) + 2

928 = 892 + (9*8)/2

928 = 829 + (92+8) + (9-2-8)

928 = 298 + 289 + 9*2*8 + 2*98 - (9-2-8)

928*829 = 769312 (all different digits). (Next number with this property?)

928*(9*2*8) = 133632.

928*(9+2+8) = 17632. (both end with same three digits) (Do other numbers have this property?)

928 - (9*2*8) = 784 = 28^2 (Note: 28 is in the string 928). (What's another number like this?)

928 is divisible by 8 and 29 (uses all the digits in 928). (Are there other numbers with this property?)

2^928, 3^928, 5^928, 7^928, and 11^928 all have even digit sums. (What other numbers are like this?)

928^4 + 1 is prime and (928+2)^4 + 1 is prime. (What's the next number with the same property?)

9281 and 9283 are prime but 9287 and 9289 are composite. (What are other numbers satisfying these conditions?)

271st Day of the Year.

271 = 27 + 71 + 21 + 127 + (2+17) - (2-7-1)

271 = 17 + 72 + 12 + 172 - 2

271 = 27*1 + 2*71 + 12*7 + (21-7) + (7-2-1)

271 - 1 = 98 + 76 + 54 + 32 + 10

271^4 does not contain a 2, 7 or 1 (disregarding the obvious ones digit). (Any other numbers like this?)

271 = 223 + 48 (48th prime plus 48)

The next prime is 277. The number 271277 is also prime (concatenation). (What other consecutive primes share this property?)

Inserting a 3 in "271" maintains a prime (i.e. 2713, 2731, 2371, and 3271 are prime). (What is the next number with this property?)

271 is a divisor of 99999 (and, namely, 9999999999).

271 is the largest prime factor of 11111.

271 = (2+7+1)^3 - (2+7)^3 = 10^3 - 9^3

The first 100 primes have 271 digits (when counted).

271 is prime and is the concatenation of two primes (2 and 71). (What other primes have this property?)

271 = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 (11 consecutive primes)

September 29th (272nd Day of the Non-Leap Year)

Sep 29, 9/29.

929 = 992 - 9*(9-2)

929 = 299 + 29*9 + 9*(29+2+9+9/9)

929 = 92 + 29 + 99 - (9-2-9)*(299) + (92+9) + (9+9/9)

929 = 92*(9+2) + (92-9)

929 = 92*(9-2) + (9+9+9/9)*(29-2*(9-2))

929 = 92*9 + 9*29 + 9*2*9 + (9-2-9)

929 = 29*(2+9) + (9+9/9)*(92-(29+2))

929 = 29*2*9 + (2+9)*(29+(9-9/9))

929 = 99*2 + 299 + (9+2+9)*(9+2+9) + 29 + (2+9/9)

929 = (9+2+9)*(29+2*9) - (29-2*9)

929 = (9*2+9)*(29+(9-2-2)) + (29-2*9)

929^2 does not contain a 9 or a 2 and has all different digits. (What's the next number with the same property?)

929*299 = 277771 (a run of four 7s). (What other numbers share a similar property? (doesn't need to be a 7))

929 - (9*2*9) = 767 (a palindrome, as well). (What are other numbers with this property?)

929 + (9*2*9) = 1091 is also prime. (What's the next number with this property?)

929 - (9+2+9) = 909.

929 + (9+2+9) = 949. (both palindromes) (What other numbers have this property?)

Today is also the 272nd Day of the Year. 272 and 929 are both palindromes. (Does this happen for another day of the year?)

929 is prime.

9^929 + 2 is also prime. (Note: the "9^" and the "+2" use digits in 929). (What other numbers are like this?)

929 is the largest prime factor of 2*3*5*7*11*13*17*19 - 1.

929 = 28^2 + 12^2 + 1^2 (Note: 28+12+1 = 41 is also prime).

929 = 30^2 + 5^2 + 2^2 (Note: 30+5+2 = 37 is also prime).

929 = 27^2 + 14^2 + 2^2 (Note: 27+14+2 = 43 is also prime).

929 = 27^2 + 10^2 + 10^2 (Note: 27+10+10 = 47 is also prime).

(Find more equations like this but instead of 3 perfect squares, can you find one with only 2? 4? Be sure they have the same property mentioned above.)

929 is a palindromic prime where a single digit is sandwiched between a string of 9s. The first one was 919. (What is the next number like this? Hint: Over 1,000,000,000!)

929 = 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 (9 consecutive primes).

272nd Day of the Year.

272 = 722 - 72*2 - 7*22 - (2+2)*(27+2+7+2)

272 = 27*(2+7) + (27+2)

272 = 27*(7-2) + (72+(7-2)*(7+2^2+2))

272 = 72*(7-2) - (7+2+2)*(7+2/2)

272 = 27 + 72 + 22 + 22*7 + (2-7+2)

272 = 227 - (2-7)*(7+2)

272 = 72*(7+2) - 227 - (2*72+(7-2))

272 = 2*72 + 27*2 + (2+72)

272 = (2+7+2)*(2*7*2) - (72/2)

272 = 27*2*7

272*(2+7+2) = 2992 (a palindrome). (What other numbers are like this?)

272*(2*7*2) contains a 3-digit palindrome. (What other numbers have this property?)

272 + (2*7*2) = 300 (multiple of 100). (What is the next number with this property?)

272^3 begins with "2012". (What other numbers have the same property?--instead of 2012, is there a 2013? 2014? Or another notable year?)

272^4 contains two 2s and one 7. (Are there other numbers that share this property?)

272 = 2*2*2*2*17

272*(2+2+2+2+17) = 6800 (multiple of 100).

272*(2+2+2+2+1+7) = 4352 (four consecutive numbers). (Are other numbers like this?)

272 = 16*17

272*(16+17) = 8976 (four consecutive numbers). (Do other numbers have this property?)

272^4 + 1 is prime.

10^272 - 9 is prime.

272^10 + 272^9 + 272^8 + 272^7 + 272^6 + 272^5 + 272^4 + 272^3 + 272^2 + 272 + 1 is prime.

(What other numbers have one, two, or all three of these properties?)

There are 272 3-digit numbers requiring 5 positive cubes in their representation as a sum of cubes.

272 = 4^4 - 2^4 (only one way to represent this number as a^4 - b^4). (What's the next number like this?)

272 = 61 + 67 + 71 + 73 (four consecutive primes).

September 30th (273rd Day of the Non-Leap Year)

Sep 30, 9/30.

930 = 93 + 30 + 90 + 309 + 390 + (3+9+0) - (3-9-0)

930 = 39 + 9*30 + (9*3)*(9+3+0) + 309 - (9+3+0)

930 = 390 + 309 + 39 + 93 + (9*3) + (9-3-0)*(9+3+0)

930 = 903 - 9*3

309 = 9*30 + (9+30)

930^2 is all composite digits. (What's the next number like this?)

930 = 2*3*5*31.

2+3+5+31 = 41.

2+3+5+3+1 = 14. (reversal of 41). (Do other numbers have the same property?)

There are 930 primes between 40,000 and 50,000.

If 930 = n, 30*n/(30-n) is an integer. This is the largest number with such property. (What's the previous number with this property?)

273rd Day of the Year.

273 = 27 + 73 + 23 + 2*73 + (7-3)

273 = 37 + 72 + 32 + 27*3 + 3*(2*7+3)

273 = 237 + (27+3) + (2+7-3)

273 = 27*3 + 2*73 - (2-7+3)*(23)

273 = 2*7*3 + (23*7) + (2+7+3) + (32+7) - (2-(7*3))

273*(2*7*3) = 11466 (starts/ends with repdigit). (Do other numbers share this property?)

273*(2+7+3) has 2,7, and 3 in its decimal representation. (Are there other numbers with this property?)

273 = 2*7*13

2+7+13 = 23 (digits are in 273). (Do other numbers have this property?)

273 = 4^4 + 2^4 + 1^4 = (2^2)^4 + (2^1)^4 + (2^0)^4

273^2 + 2 and 273^2 - 2 are both prime. (What other numbers are like this?)

273 is a divisor of 999,999 (and 111,111). (What's the next number that shares this property?)

10^273 - 9^272 is prime. (What other numbers have this property?)

There are 273 perfect squares less than 42^3.

2730 + 1 and 2730 - 1 are twin primes. (What is the next number with this property?)

There are only 5 groups of order 273. (Can you name one?)

|10^273 - 273^10| is a prime number. (Only four like this are known. The first two are 3 and 9. What's the last one?)

273 is a repdigit in base 9 (333), base 16 (111), base 20 (DD), base 38 (77), and base 90 (33).

273 is a palindrome in base 2 (100010001), and base 4 (10101) along with the repdigit bases.

(What other numbers have similar properties?)

The zero on the Celsius scale is (to the nearest whole number) 273 K. Hence, absolute zero (0 K) is around -273 deg C.

273 K is the triple point of water (all three of its states coexist).

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