An Identity of Fillipi Related to Fermat-Type Equations

Abstract

We give a structural proof of an algebraic identity proposed by Fillipi in 2014. For every integer $n\ge 2$, the identity represents an explicit multiple of $x^n+y^n-z^n$ as ${\mathcal{A}}^2+{\mathcal{B}}^2-{\mathcal{C}}^2$. We identify this expression with a determinant arising from a weighted Gram matrix and show that Fillipi's particular fourth-power terms are obtained from a simpler identity by one polynomial column transformation. The same transformation yields an explicit polynomial rank-one factorisation modulo $x^n+y^n-z^n$. Thus the associated Pythagorean relation is a universal polynomial consequence of the Fermat-type equation and supplies no additional algebraic solvability condition.

6 pages. Substantially revised version with a structural derivation via a weighted Gram determinant, a polynomial column transformation, and an explicit rank-one factorization on the Fermat hypersurface