A note on Fermat's Last Theorem for $n=4$

TL;DR

This paper explores a specific Diophantine equation related to Fermat's Last Theorem for n=4, presenting an infinite sequence of solutions to a modified version of the theorem.

Contribution

It introduces an infinite sequence of integer solutions to a related quartic equation, offering a new perspective on Fermat's Last Theorem for n=4.

Findings

Identified an infinite sequence of solutions to x^4 + y^4 - 8 = z^2

Provides a quantitative perspective on Fermat's Last Theorem for n=4

Suggests possible extensions to related Diophantine equations

Abstract

Fermat's Last theorem (FLT) famously states that the equation $x^n+y^n=z^n$ has no solution in positive integers $x, y, z$ for any integer exponent $n>2$. But does this theorem have a quantitative version? Upon initial investigation we discovered an infinite sequence of integers $(x_n, y_n, z_n)$ with $x_n^4+y_n^4-8=z_n^2$.