Random Thue and Fermat equations

TL;DR

This paper investigates the solvability of certain Thue and Fermat equations, showing that under the $abc$-conjecture, most such equations violate the Hasse principle, indicating a discrepancy between local and global solutions.

Contribution

It demonstrates that assuming the $abc$-conjecture, almost all locally soluble Thue and Fermat equations of specified degrees violate the Hasse principle, revealing new insights into their solvability.

Findings

Most Thue equations of degree ≥ 3 violate the Hasse principle.

Most Fermat equations of degree ≥ 6 violate the Hasse principle.

Results depend on the assumption of the $abc$-conjecture.

Abstract

We consider Thue equations of the form $ax^k+by^k = 1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $ax^k+by^k+cz^k = 0$ of degree at least six.