Rational right triangles and the Congruent Number Problem

TL;DR

This paper explores the relationships between rational right triangles and congruent numbers using geometric methods, conic sections, and recursive algorithms, revealing new connections with Fibonacci, Lucas, and Chebyshev polynomials.

Contribution

It introduces novel geometric and algebraic methods linking congruent numbers to rational triangles, including recursive constructions and polynomial relationships, advancing understanding of the congruent number problem.

Findings

Relationship between Cassini ovals and congruent numbers

Infinite trees of rational triangles related to congruent numbers

Connections between semiperimeters of Brahmagupta triangles and Chebyshev polynomials

Abstract

From Euclid's fundamental formula for the Pythagorean triples we define the rational triples relating certain congruent numbers by an identity and explore their relationships. We introduce two geometric methods relating the congruent number problem to pairs of conic sections. We show a relationship between the Cassini ovals and the congruent number problem. By the tangent method we define a set of rational triangles from an initial solution for a congruent number. We define the prime footprint equations for right triangles for certain congruent numbers. By the unseen recurrence we define infinite trees of rational triangles and congruent numbers. Congruent number families are defined related to the Fibonacci/Lucas numbers and the Chebyshev polynomials. We show that the semiperimeter of the Brahmagupta triangles are congruent numbers in function of the Chebyshev polynomials of the first kind. We present a naive recursive algorithm to solve the problem posed by Fermat for triangles with $a+b=\square$ and $c=\square$