Double exponential lower bounds for possible solutions in the Second Case of the Fermat Last Theorem

TL;DR

This paper improves lower bounds for solutions of the second case of Fermat's Last Theorem using local power series methods, aiming to complement computational verifications and advance understanding of solution bounds.

Contribution

It provides a strengthened lower bound for FLT2 solutions, enhancing previous bounds to better utilize computational results and theoretical inequalities.

Findings

Enhanced lower bounds for FLT2 solutions

Connection to effective abc inequality and Mochizuki's work

Potential to leverage computational verifications more effectively

Abstract

In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper \cite{MFHMP} based on Mochizuki's \cite{Mo}[IUT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2, which is necessary in order to take advantage of the best upper bounds for primes $p$ for which it was verified on a computer that FLT2 has no solutions.