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The twin composites may be defined by analogy with the twin primes as pairs of composite numbers (n,n+2). Since all even number are trivially twin composites, it is natural ...

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers k such that k·2^n+1 is composite for every n>=1. Numbers k with this property are called ...

An integer is k-smooth if it has no prime factors >k. The following table gives the first few k-smooth numbers for small k. Berndt (1994, p. 52) called the 7-smooth numbers ...

In determinant expansion by minors, the minimal number of transpositions of adjacent columns in a square matrix needed to turn the matrix representing a permutation of ...

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 ...

The Pell numbers are the numbers obtained by the U_ns in the Lucas sequence with P=2 and Q=-1. They correspond to the Pell polynomial P_n(x) and Fibonacci polynomial F_n(x) ...

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers n. Pick values of r given by r=|_sqrt(n)_|+k, (1) where k=1, 2, ...

A Dyck path is a staircase walk from (0,0) to (n,n) that lies strictly below (but may touch) the diagonal y=x. The number of Dyck paths of order n is given by the Catalan ...

The (lower) irredundance number ir(G) of a graph G is the minimum size of a maximal irredundant set of vertices in G. The upper irredundance number is defined as the maximum ...

Let P(N) denote the number of primes of the form n^2+1 for 1<=n<=N, then P(N)∼0.68641li(N), (1) where li(N) is the logarithmic integral (Shanks 1960, pp. 321-332). Let Q(N) ...