A Pythagorean quadruple is a set of positive integers a, b, c, and d that satisfy a^2 + b^2 + c^2 = d^2. For positive even a and b, there exist such integers c and d; for positive odd a and b, no such integers exist. Examples of primitive Pythagorean quadruples include (1, 2, 2, 3), (2, 3, 6, 7), (4, 4, 7, 9), (1, 4, 8, 9), (6, 6, 7, 11), and (2, 6, 9, 11). Oliverio gives the following generalization of this result. Let S = (a_1, ..., a_(n - 2)), where a_i are integers, and let T be the number of odd integers in S. Then iff T not congruent 2 (mod 4), there exist integers a_(n - 1) and a_n such that a_1^2 + a_2^2 + ... + a_(n - 1)^2 = a_n^2.

Diophantine equation--4th powers | Euler brick | Pythagorean triple | sum of squares function