An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares

TL;DR

This paper proves the existence of infinitely many primitive Heron triangles with two sides as perfect squares using elliptic curve theory, revealing a new infinite family of such triangles and suggesting all sides could be perfect squares.

Contribution

It introduces a novel method of nesting elliptic curves to establish an infinite family of primitive Heron triangles with two perfect square sides.

Findings

Existence of infinitely many such triangles proven.

A rational point on a nested elliptic curve corresponds to these triangles.

Potential for all sides to be perfect squares in such triangles.

Abstract

A primitive Heron triangle is a triangle with integral sides and integral area where the greatest common divisor of the lengths of its sides is $1$. By utilizing the theory of elliptic curves, we prove that there exist infinitely many primitive Heron triangles with two sides being perfect squares. In this process, we nest one elliptic curve into another and find a surprising rational point. All the Heron triangles corresponding to this rational point are primitive. This result would imply the possible existence of infinitely many primitive Heron triangles with all three sides being perfect squares.