Primitive norm extraction and a signed descent for the cubic Fermat equation

Abstract

We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\mathbb{Z}[\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\mathbb{Z}[\sqrt{-3}]$ lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform $2K(K^2+3M^2)=C^3$ into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives $|C^{\prime }|\le |C|/39$. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.

arXiv comment10 pages