Polygonal triples and the double ruling of a hyperboloid

Abstract

For a given positive integer k ≠ 4, let P _k,n denote the n-th k-gonal number. We study k-gonal triples (a, b, c) satisfying

A k-gonal triple corresponds to a rational point on the rectangular hyperboloid x ² + y ² = z ² + 1. The simple observation that this is a doubly ruled surface leads to an easy, complete determination of k-gonal triples. The main result is expressed in terms of primitive Pythagorean triples. We also establish a correspondence between (2h + 1)-gonal triples and 4h-gonal triples.

Keywords:

• polygonal triples
• rectangular hyperboloid
• primitive Pythagorean triples