A Quartic Identity Related to Fermat-Type Equations

TL;DR

The paper presents a simple algebraic identity that transforms Fermat-type equations into Pythagorean triples, providing a new perspective on these classical equations.

Contribution

It offers a concise proof of an algebraic identity linking Fermat equations to quadratic forms, enabling polynomial expressions of solutions.

Findings

Expresses $(x^n + y^n - z^n)$ as a quadratic form after multiplication.

Transforms hypothetical solutions of $x^n + y^n = z^n$ into explicit Pythagorean triples.

Provides a new algebraic tool for analyzing Fermat-type equations.

Abstract

We provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit factor. Consequently, any hypothetical solution of $x^n+y^n=z^n$ yields a Pythagorean triple $(\mathcal A,\mathcal B,\mathcal C)$ consisting of explicit polynomial expressions in $x,y,z$.