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

Yangcheng Li

TL;DR

The paper proves the existence of infinitely many primitive Heron triangles with two sides being perfect squares by embedding a Diophantine condition into an elliptic-curve framework. It introduces a chain of transformations culminating in curves ${\mathcal E}_{A}$ (and variants) whose positive rank yields infinite rational points corresponding to such triangles, with explicit $A=4$ instances providing concrete primitive examples. A primitivity analysis ensures gcd conditions hold to produce primitive triangles, and the authors extend the method to primitive isosceles cases via a related curve ${\mathcal E}_{A}'$. The work suggests the potential existence of infinitely many primitive Heron triangles with all three sides squares and showcases the power of elliptic-curve techniques in Diophantine geometry.

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.

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

TL;DR

The paper proves the existence of infinitely many primitive Heron triangles with two sides being perfect squares by embedding a Diophantine condition into an elliptic-curve framework. It introduces a chain of transformations culminating in curves (and variants) whose positive rank yields infinite rational points corresponding to such triangles, with explicit instances providing concrete primitive examples. A primitivity analysis ensures gcd conditions hold to produce primitive triangles, and the authors extend the method to primitive isosceles cases via a related curve . The work suggests the potential existence of infinitely many primitive Heron triangles with all three sides squares and showcases the power of elliptic-curve techniques in Diophantine geometry.

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 . 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.
Paper Structure (2 sections, 3 theorems, 45 equations, 1 figure)

This paper contains 2 sections, 3 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proofs of the theorems

Key Result

Theorem 1.1

Let $A$ be a fixed positive rational number. If the elliptic curve has a positive rank, then there are infinitely many scalene Heron triangles with two sides being perfect squares.

Figures (1)

  • Figure 1: A Heron triangle with two sides being perfect squares.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem 1.1.
  • Example 2.1
  • proof : Proof of Theorem 1.2.
  • Example 2.2
  • proof : Proof of Theorem 1.3.
  • Example 2.3