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An NSW number (named after Newman, Shanks, and Williams) is an integer m that solves the Diophantine equation 2n^2=m^2+1. (1) In other words, the NSW numbers m index the ...

A figurate number Te_n of the form Te_n = sum_(k=1)^(n)T_k (1) = 1/6n(n+1)(n+2) (2) = (n+2; 3), (3) where T_k is the kth triangular number and (n; m) is a binomial ...

A number which is simultaneously square and triangular. Let T_n denote the nth triangular number and S_m the mth square number, then a number which is both triangular and ...

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

There are two kinds of power sums commonly considered. The first is the sum of pth powers of a set of n variables x_k, S_p(x_1,...,x_n)=sum_(k=1)^nx_k^p, (1) and the second ...

A number h which satisfies the conditions of the congruum problem: x^2+h=a^2 and x^2-h=b^2, where x,h,a,b are integers. The list of congrua is given by 24, 96, 120, 240, 336, ...

For every k>1, there exist only finite many pairs of powers (p,p^') with p and p^' natural numbers and k=p^'-p.

Given a Pythagorean triple (a,b,c), the fractions a/b and b/a are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of ...

A Pythagorean triangle is a right triangle with integer side lengths (i.e., whose side lengths (a,b,c) form a Pythagorean triple). A Pythagorean triangle with GCD(a,b,c)=1 is ...

A conjecture that, as proved by Parshin (1968), implies the Mordell conjecture.