Fermat’s Last Theorem and Catalan’s conjecture in weak exponential arithmetics
This is a joint work with Vitezslav Kala.
Wiles’s proof of Fermat’s Last Theorem (FLT) has stimulated a lively discussion on how much is actually needed for the proof.
Despite the fact that the original proof uses set-theoretical assumptions unprovable in Zermelo-Fraenkel set theory with axiom of choice (ZFC) (namely, the existence of Grothendieck universes), it is widely believed that
certainly much less than ZFC is used in principle, probably nothing beyond Peano arithmetic, and perhaps much less than that.
I will start with a brief summary of existing positive and negative results on provability of FLT in various arithmetical theories.
In this talk, we will consider structures and theories in the language L=(0,1,+,x,<,e), where the symbol e is intended for a (partial or total) binary exponential. We show that Fermat's Last Theorem for e (i.e. the statement "e(a,n)+e(b,n)=e(c,n) has no non-zero solution for n>2″) is not provable in the L-theory Th(N)+Exp, where Th(N) stands for the complete theory of the standard model N=(N,0,1,+,x,<) and Exp is a natural set of axioms for e (consisting mostly of elementary identities). On the other hand, under the assumption of ABC conjecture (in the standard model), we show that the Catalan conjecture for e is provable in Th(N)+Exp (even in a weaker theory). This gives an interesting separation of strengths of these two diophantine problems. 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". Slides from this talk.
Petr Glivický is a Researcher at Charles University in Prague, in the Department of Theoretical Computer Science and Mathematical Logic, where he received his doctorate in 2013 as a student of Josef Mlček. His research interests include model theory, Peano Arithmetic, and non-standard analysis.