Fermat's last theorem and Bezout's theorem in GCD domains (Q1196796)

The author considers the extent to which cases of Fermat's ``last theorem'' may be proved in discretely ordered rings which have some weak form of induction. \textit{J. C. Shepherdson} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 12, 79-86 (1964; Zbl 0132.247)] constructed models of open induction in which \(x^ n+y^ n=z^ n\) has solutions for every natural number \(n\). Here the author considers models of open induction in which every pair of elements has a gcd. More generally he considers \(\mathbb{Z}\)- rings which have gcd's (a \(\mathbb{Z}\)-ring is a discretely ordered ring which, when factored by the ideal generated by \(n\) (for any \(n)\), is isomorphic to the integers modulo \(n)\). He shows that Germain's theorem holds in such rings and deduces that if \(p\) is an odd prime not congruent to 1 modulo 8 then case 1 of Fermat's last theorem holds in the ring for \(n=2p\) (that is, for any solution to \(x^ n+y^ n=z^ n\), \(n\) must divide \(xyz)\). --- The author also proves that in any model of the stronger system of bounded existential induction [\textit{G. Wilmers}, J. Symb. Logic 50, 72-90 (1985; Zbl 0634.03029)] Fermat's last theorem holds for \(n=2,6,10\). The paper is accessible and clearly written.