An octic diophantine equation and related families of elliptic curves

TL;DR

This paper presents parametric solutions to a specific octic Diophantine equation, leading to infinitely many equiareal triangles with sides as perfect squares, and constructs elliptic curve families of rank 5 over rational functions.

Contribution

It introduces new parametric solutions to an octic Diophantine equation and explores associated elliptic curve families of high rank, providing explicit generators.

Findings

Infinite solutions for equiareal triangles with square sides.

Construction of elliptic curves of rank 5 over rac{ ext{Q}}{ ext{t}}.

Explicit generators for the elliptic curve family.

Abstract

We obtain two parametric solutions of the diophantine equation $\phi(x_1, x_2, x_3)=\phi(y_1, y_2, y_3)$ where $\phi(x_1, x_2, x_3)$ is the octic form defined by $\phi(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.