Congruent Numbers Via the Pell Equation and its Analogous Counterpart

TL;DR

This paper explores the relationship between congruent numbers, Pythagorean triples, and Pell equations, introducing polynomials and demonstrating how these connections generate congruent numbers with multiple prime factors.

Contribution

It presents new polynomial constructions and links between Pythagorean triples, Pell equations, and congruent numbers, expanding understanding of their interrelations.

Findings

Polynomials with values as congruent numbers introduced

Connections established between Pythagorean triples and Pell equations

Generation of congruent numbers with arbitrarily many prime factors

Abstract

The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections between Pythagorean triples and the Pell equation $x^2-dy^2=1$ plus its analogous counterpart $x^2-dy^2=-1$ which give rise to congruent numbers n with arbitrarily many prime factors.