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If p^k is the highest power of a prime p dividing the order of a finite group G, then a subgroup of G of order p^k is called a Sylow p-subgroup of G.

Let p be an irregular prime, and let P=rp+1 be a prime with P<p^2-p. Also let t be an integer such that t^3≢1 (mod P). For an irregular pair (p,2k), form the product ...

An oriented surface for which every point belongs to a Wiedersehen pair. Proof of the Blaschke conjecture established that the only Wiedersehen surfaces are the standard ...

Any composite number n with p|(n/p-1) for all prime divisors p of n. n is a Giuga number iff sum_(k=1)^(n-1)k^(phi(n))=-1 (mod n) (1) where phi is the totient function and ...

Two nonisomorphic graphs that have equal resistance spectra (i.e., multisets of resistance distances) are said to be resistance-equivalent. All nonisomorphic simple graphs on ...

van der Waerden's theorem is a theorem about the existence of arithmetic progressions in sets. The theorem can be stated in four equivalent forms. 1. If N=C_1 union C_2 union ...

7 7 6 6 3 1; 6 5 4 2 ; 3 3 ; 2 A descending plane partition of order n is a two-dimensional array (possibly empty) of positive integers less than or equal to n such that the ...

Let a^p+b^p=c^p be a solution to Fermat's last theorem. Then the corresponding Frey curve is y^2=x(x-a^p)(x+b^p). (1) Ribet (1990a) showed that such curves cannot be modular, ...

Honaker's problem asks for all consecutive prime number triples (p,q,r) with p<q<r such that p|(qr+1). Caldwell and Cheng (2005) showed that the only Honaker triplets for ...

Mann's theorem is a theorem widely circulated as the "alpha-beta conjecture" that was subsequently proven by Mann (1942). It states that if A and B are sets of integers each ...