Abstract

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions of models of arithmetical theories (in the language ) 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. We construct a model and a substructure with e total and (Presburger arithmetic) such that in both and Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples. 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 (even in a weaker theory) and thus holds in and. Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in ) in “coprimality for e”.