Department of
Applied
Mathematics

(Un)provability of Fermat's last theorem and Catalan's conjecture in formal arithmetics

Petr Glivický

Abstract

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." (McLarty)

In this talk, I will present a joint work with V. Kala. We studied (un)provabiliy of FLT and Catalan's conjecture in arithmetical theories with weak exponentiation, i.e. in theories in the language L=(0,1,+,x,exp,<) where the (0,1,+,x,<)-fragment is usually very strong (often even the complete theory Th(N) of natural numbers in that language) but the exponentiation satisfies only basic arithmetical properties and not much of induction. In such theories, Diophantine problems such as FLT or Catalan's conjecture, are formalized using the exponentiation exp instead of the exponentiation definable in the (0,1,+,x,<)-fragment.

I will present a natural basic set of axioms Exp for exponentiation (consisting mostly of elementary identities) and show that the theory T=Th(N)+Exp is strong enough to prove Catalan's conjecture, while FLT is still unprovable in T. This gives an interesting separation of strengths of the two famous Diophantine problems. Nevertheless, I show that by adding just one more axiom for exponentiation (the, so called, "coprimality" of exp) the theory becomes strong enough to prove FLT, i.e. FLT is provable in T+"coprimality". (Of course, in the proof of this, we use the Wiles's result too.)