Notes on the Diophantine Equation A^4+aB^4=C^4+aD^4

TL;DR

This paper introduces a new formulation of a specific Diophantine equation, derives multiple solutions including a historical parametric solution, and presents numerous examples and triplet solutions.

Contribution

It provides a novel formulation and a comprehensive set of parametric and semi-parametric solutions for the equation, including previously unpublished results.

Findings

Derived several parametric solutions

Presented numerous explicit examples

Included eight triplet solutions

Abstract

A new formulation of the subject equation is presented. Several parametric and semi-parametric solutions are derived. The parametric solution for a=-1 was originally presented in 1972, but never published. A computer-generated version was found by Zajta and published in 1983. The pencil-and-paper version is presented for the record, along with many other examples, including eight coincidental triplet solutions.