The set of exponents, for which Fermat’s last theorem is true, has density one. (English) Zbl 0565.10016

In his highly important papers [Invent. Math. 73, 349-366 (1983), ibid. 75, 381 (1984)], G. Faltings proved Mordell’s conjecture, among other striking theorems. In the case of Fermat’s equation, it says that for every \(n>3\), there exist at most finitely many triples of relatively prime non-zero integers (x,y,z) such that \(x^ n+y^ n=z^ n\); of course, if \(n=3\) it was known that there is none. The result of Faltings is however ineffective: no bound for the number or size of solutions is indicated. M. Filaseta [ibid. 6, 31-32 (1984; Zbl 0533.10010)] proved in 1984 several interesting corollaries of Faltings’ theorem applied to Fermat’s equation. In particular, for every integer \(r\geq 3\) there exists N(r)\(\geq 1\) such that if \(m>N(r)\) and \(n=mr\) then \(X^ n+Y^ n=Z^ n\) has only trivial solutions; N(r) is not effectively computable.
In the present paper, the author obtains the following very neat consequence of Faltings’ and Filaseta’s results. For every \(N>1\) let \(\nu\) (N) be the number of integers n, \(1\leq n\leq N\), such that \(X^ n+Y^ n=Z^ n\) has only trivial solutions. Then it is shown: \(\lim_{N\to \infty}\nu (N)/N=1\). The method of proof is elementary and elegant.
The present result improves on a previous one by N. C. Ankeny and P. Erdős [Am. J. Math. 76, 488-496 (1954; Zbl 0056.035)], who arrived at this theorem, with \(\nu_ 1(N)\) in place of \(\nu\) (N); here \(\nu_ 1(N)\) denotes the number of integers n, \(1\leq n\leq N\), such that if x,y,z are non-zero integers and \(x^ n+y^ n=z^ n\) then \(\gcd (n,xyz)>1\). The theorem holds also for any generalized Fermat’s equation \(aX^ n+bY^ n=cZ^ n\) with a,b,c non-zero integers and, in case \(\pm a\pm b=c\), \((\pm 1,\pm 1,1)\) is to be considered also a trivial solution.
It should be noted that, independently and just about the same time, D. R. Heath-Brown published a paper containing the same result as in this paper, the proofs being basically the same [see Bull. Lond. Math. Soc. 17, 15-16 (1985; Zbl 0546.10012)]. Finally, S. Wagon gave an even simpler proof of this result [Math. Intell. 7, 72-76 (1985; Zbl 0566.10008)].
Reviewer: P.Ribenboim


11D41 Higher degree equations; Fermat’s equation