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Propositions of Mine (and Others...)

1! + 2! + 3! + ... + N! is not a perfect square or a perfect cube for N > 3 (Confirmed for N < 10,000).

(Credit: Derek Orr)

(Perfect square proven on http://asgarli.wordpress.com/2012/10/06/sum-of-first-n-factorials/)

(Perfect cube proven by Derek Orr)

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Prove that for prime numbers P, P + DigitSum(P) and P - DigitSum(P) both cannot return perfect squares unless P = 17, where DigitSum() is the digital sum of P. However, P + DigitSum(P) and P - DigitSum(P) both cannot return perfect cubes for any value of P (Both confirmed for P < 1,000,000).

(Credit: Derek Orr)

(Proven by Derek Orr)

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Prove that for prime numbers P, P + DigitProd(P) and P - DigitProd(P) both cannot return perfect squares, where DigitProd() is the digital product of P. Further, P + DigitProd(P) and P - DigitProd(P) both cannot return perfect cubes (Both confirmed for P < 1,000,000).

(Credit: Derek Orr)

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1! + 2! + 3! + ... + N! is prime only when N = 2 (Confirmed for N < 1,000).

(Credit: Derek Orr)

(Proven by Derek Orr)

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If N is a perfect cube, then the sum of its proper divisors is not a perfect cube.

(Credit: Derek Orr)

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If the sum and product of the proper divisors of N are palindromes, then N is either a palindrome or a perfect square.

(Credit: Derek Orr)

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Prove that for any positive integer N and natural number B,

IteratedDigitSum(N^B) = IteratedDigitSum[IteratedDigitSum(N)^B]

Ex.

If N = 548 and B = 3.

IteratedDigitSum(548^3) = IteratedDigitSum(164566592) = 8.

IteratedDigitSum[IteratedDigitSum(548)^3] = IteratedDigitSum(8^3) = IteratedDigitSum(512) = 8.

This claim says that these values will always be equal for any integer N > 0 and any integer B > 0.

Not sure if this is related to the IteratedDigitSum problem below.

(Credit: Derek Orr)

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Prove that if N*reverse(N) is a perfect cube, then N must also be a perfect cube. (The reverse(N) flips the digits. Ex. reverse(4321) = 1234.)

(Credit: Derek Orr)

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Prove that if N is a perfect cube > 8, then its DigitSum and DigitProduct cannot both be perfect cubes.

(Credit: Derek Orr)

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Prove for N > 2, if N is prime, then N^N + 1 or N^N - 1 cannot be prime.

(Credit: Derek Orr)

(Proven by Chris J Robot @chrisjrobot)

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Prove for N > 2, if sigma(N) is prime, then N is a perfect square (sigma(N) denotes the sum of the divisors of N)

(Credit: Derek Orr)

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Prove every semiprime is a deficient number or a perfect number.

(Credit: Derek Orr)

(Proven by Derek Orr and Chris J Robot @chrisjrobot)

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Given two integers A and B.

Prove that there exists two other integers m,n such that GCD(A,B) = mA + nB.

(Credit: Derek Orr)

(Proven by Derek Orr)

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Let the iterated digit sum of the number A = x.

Let the iterated digit sum of the number B = y.

And, A*B = C.

Prove that as long as the iterated digit sum of A remains x and the iterated digit sum of B remains y, then C will always have the same iterated digit sum.

Ex.

Iterated digit sum of 754 = 7+5+4 = 16 => 1+6 = 7.

Iterated digit sum of 614 = 6+1+4 = 11 => 1+1 = 2.

754*614 = 462956.

Iterated digit sum of 462956 = 4+6+2+9+5+6 = 32 => 3+2 = 5.

This claim says that any two numbers (one with iterated digit sum of 7 and the other with iterated digit sum of 2) multiplied together will always return a number with the iterated digit sum 5 (20*16 =320, 7*2, 11*25, etc.).

(Credit: Derek Orr)

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Prove that cbrt(18 - sqrt(325)) + cbrt(18 + sqrt(325)) = 3.

(Credit: @MathUpdate)

(Proven by Derek Orr)

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Prove that 767, 76767, 7676767, etc. are all composite.

(Credit: Dave Radcliffe @daveinstpaul)

(Proven by Derek Orr)

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Prove that 343, 34343, 3434343, etc. are all composite.

(Credit: Derek Orr)

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Prove that 717, 71717, 7171717, etc. are all composite.

(Credit: Derek Orr)

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For any four digit multiple of 99, abcd, prove that bcda, cdab, and dabc are also multiples of 99.

(Credit: Derek Orr)

(Proven by Derek Orr)

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Let p,q,r,s be prime and 7 =< p < q < r < s. If q - p = s - r = 2^k (for some positive integer k), then r - p = s - q = 6n (where n is another positive integer)

(Credit: @MathUpdate and Derek Orr)

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Prove that a number with 9 divisors is a perfect square.

(Credit: Derek Orr)

(Proven by Derek Orr)

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Prove that if a^2 + b^2 = c^2 and a,b,c are coprime, then c is odd.

(Credit: Derek Orr)

(Proven by Derek Orr)

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Prove that in the (x,y) plane, for odd integers A,B,C, the line Ax + By + C = 0 cannot intersect the parabola y = x^2 at a rational point.

(Credit: @MathWeekly)

(Proven by Derek Orr)

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Find all the real solutions for: 4x^2 - 40[x] + 51 = 0 where [x] denotes the floor function of x.

(Credit: @MathWeekly)

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Prove that every positive integer has a multiple whose decimal representation involves the sequence: 20102011.

(Credit: @MathWeekly)

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Prove that if n is an integer greater than 1, then 2^n - 1 is not divisible by n.

(Credit: Dave Radcliffe @daveinstpaul)

(Proven by Derek Orr)

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Simplify this nested radical: sqrt(1+2*sqrt(1+3*sqrt(1+4*sqrt(1+...))))

(Credit: Dave Radcliffe @daveinstpaul)

(Proven by Derek Orr)

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If f(x)^2 = x + f(x+1) and f(1) = A (for some real number A), what is f(x)?

(Credit: Derek Orr)

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If f(x)^2 = x + f(2x) and f(1) = B (for some real number B), what is f(x)?

(Credit: Derek Orr)

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Take the sequence x_n. Let x_1 = x_1.

x_(n+1) := (x_n + a)/(x_n + b), for all natural numbers a,b.

Prove lim(x_n) is the positive solution to the quadratic equation:

x^2 + (b-1)x - a = 0.

(Special Case: Let x_(n+1) := (x_n + a)/(x_n + 1). Prove lim(x_n) = sqrt(a).)

(Credit: Derek Orr)

(Proven by Derek Orr)

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Let f: R --> R be a continuous function such that f(x) = f(x^2).

Prove f is a constant function.

(Credit: Derek Orr)

(Proven by Derek Orr)