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 L = ) 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 satisfies Th(N) + Exp and a substructure with e total and A satisfies Pr ( 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 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 and. 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". (C) 2017 WILEY-VCH Verlag GmbH & Co.
KGaA, Weinheim