Solubility of Fermat equations

TL;DR

This paper investigates the solutions of Fermat equations, demonstrating that the set of coefficients allowing non-zero integer solutions is extremely sparse, with density zero.

Contribution

It establishes that for degree d > 1, the coefficients leading to solutions form a set of density zero, advancing understanding of Fermat equations' solvability.

Findings

Set of coefficients with solutions has density zero

Solutions exist only for a sparse subset of coefficients

Provides new insights into the distribution of solutions

Abstract

The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.