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

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".