Fermat's Last Theorem and Catalan's Conjecture in Weak Exponential Arithmetics

TL;DR

This paper investigates Fermat's Last Theorem and Catalan's conjecture within weak arithmetical systems with exponentiation, constructing models where these conjectures behave differently and analyzing their provability under certain assumptions.

Contribution

It introduces a universal construction of exponential functions in weak arithmetics and explores the conditions under which Fermat's Last Theorem and Catalan's conjecture hold or fail.

Findings

Constructed models violating Fermat's Last Theorem cofinally often.

Proved Catalan's conjecture for exponential functions assuming ABC conjecture.

Showed Fermat's Last Theorem is provable under ABC and coprimality assumptions.

Abstract

We study Fermat's Last Theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions (B,e) of models of arithmetical theories (in the language L=(0,1,+,x,<)) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms Exp. We construct a model (B,e) of Th(N) + Exp and a substructure (A,e) with e total and A model of Pr (Presburger arithmetic) such that in both (B,e) and (A,e) Fermat's Last Theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples a,b,c. On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in Th(N)+Exp (even in a weaker theory) and thus holds in (B,e) and (A,e). Finally, we also show that Fermat's Last Theorem for e is provable (again, under the assumption of ABC in N) in Th(N)+Exp+"coprimality for e".