Diophantine Equation $n(x^4+y^4)=z^4+w^4$

TL;DR

This paper reviews recent progress on the Diophantine equation n(x^4 + y^4) = z^4 + w^4, highlighting the discovery of infinite solutions and parametric solutions for specific values of n.

Contribution

It summarizes the development of solutions for the equation, including the use of elliptic curves and parametric methods for particular n values.

Findings

Infinite solutions for n=41,136,313,1028,1201,3281

Parametric solutions for n=17,257,626,641,706,1921

Use of elliptic curves in solving the equation

Abstract

In 2016 Izadi and Nabardi (b) showed (4-2-4) has infinitely many integer solutions. They used a specific congruent number elliptic curve.In 2019 Janfada and Nabardi,item C, showed that a necessary condition for n to have an integral solution for the above equation and gave a parametric solution.They gave the numeric solutions for n=41,136,313,1028,1201,3281. In 2020 Ajai Choudhry , Iliya Bluskov and Alexander James (a), showed that equation (4-2-4) has infinitely many parametric solutions. They gave the numeric solutions for n=17,257,626,641,706,1921.