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The totient function phi(n), also called Euler's totient function, is defined as the number of positive integers <=n that are relatively prime to (i.e., do not contain any ...

The Ackermann function is the simplest example of a well-defined total function which is computable but not primitive recursive, providing a counterexample to the belief in ...

The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The ...

The inverse cosine is the multivalued function cos^(-1)z (Zwillinger 1995, p. 465), also denoted arccosz (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; ...

The inverse hyperbolic tangent tanh^(-1)z (Zwillinger 1995, p. 481; Beyer 1987, p. 181), sometimes called the area hyperbolic tangent (Harris and Stocker 1998, p. 267), is ...

A linear recurrence equation is a recurrence equation on a sequence of numbers {x_n} expressing x_n as a first-degree polynomial in x_k with k<n. For example ...

The number M_2(n) = 1/nsum_(k=1)^(n^2)k (1) = 1/2n(n^2+1) (2) to which the n numbers in any horizontal, vertical, or main diagonal line must sum in a magic square. The first ...

A twin Pythagorean triple is a Pythagorean triple (a,b,c) for which two values are consecutive integers. By definition, twin triplets are therefore primitive triples. Of the ...

Iff p is a prime, then (p-1)!+1 is a multiple of p, that is (p-1)!=-1 (mod p). (1) This theorem was proposed by John Wilson and published by Waring (1770), although it was ...

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is ...