A Quartic Identity Related to Fermat-Type Equations
Mike Winkler, Andreas Fillipi
TL;DR
The paper presents an algebraic identity that expresses $x^n+y^n-z^n$ as a quadratic form after multiplication by an explicit factor, connecting Fermat-type equations to a quartic Pythagorean structure. It constructs an explicit triple $(\mathcal{A},\mathcal{B},\mathcal{C})$ and proves the key relation $(8rst)^2 (xyz)^{n-2}(x^n+y^n-z^n)=\mathcal{A}^2+\mathcal{B}^2-\mathcal{C}^2$ using two lemmas: a quartic Pythagorean triple and a mixed-term factorisation, via an isotropic bilinear form. Consequently, any solution of $x^n+y^n=z^n$ yields a (possibly non-primitive) Pythagorean triple, and the problem reduces to a canonical quadratic Diophantine system in variables $P,Q,R$ with $R^2=PQ$. The paper also casts this system in Euclid parameter form with a scale factor, illustrating a structured approach to analyzing Fermat-type equations through algebraic parametrisations, though solvability of the auxiliary system is left open.
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 $x,y,z$.