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Elder's theorem is a generalization of Stanley's theorem which states that the total number of occurrences of an integer k among all unordered partitions of n is equal to the ...

A k×n Latin rectangle is a k×n matrix with elements a_(ij) in {1,2,...,n} such that entries in each row and column are distinct. If k=n, the special case of a Latin square ...

There are several related theorems involving Hamiltonian cycles of graphs that are associated with Pósa. Let G be a simple graph with n graph vertices. 1. If, for every k in ...

The m×n king graph is a graph with mn vertices in which each vertex represents a square in an m×n chessboard, and each edge corresponds to a legal move by a king. The number ...

There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. ...

Wang's conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The conjecture was refuted when Berger ...

Let N^* be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n>=N^*, a complete graph K_4 of one color with ...

The sequence of Fibonacci numbers {F_n} is periodic modulo any modulus m (Wall 1960), and the period (mod m) is the known as the Pisano period pi(m) (Wrench 1969). For m=1, ...

A primefree sequence is sequence whose terms are never prime. Graham (1964) proved that there exist relatively prime positive integers a and b such that the recurrence ...

Brocard's problem asks to find the values of n for which n!+1 is a square number m^2, where n! is the factorial (Brocard 1876, 1885). The only known solutions are n=4, 5, and ...