Fermat’s Last Theorem for ”almost all” exponents. (English) Zbl 0546.10012

Heath-Brown, D. R.

Bull. Lond. Math. Soc. 17, 15-16 (1985).

Let N(x) denote the number of exponents 3\(\leq n\leq x\) for which \(u^ n+v^ n=w^ n\) has a non-trivial solution in integers. Fermat’s Last Theorem is the conjecture that \(N(x)=0\). The present paper shows that \(N(x)=o(x)\) as \(x\to\infty \). The proof is extremely simple. It uses G. Faltings’ result that there are finitely many primitive solutions (u,v,w) for each exponent \(n\geq 3\) [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)], together with the sieve of Eratosthenes.

MSC:

11D41	Higher degree equations; Fermat’s equation
11N35	Sieves

Keywords:

Faltings theorem; Mordell conjecture; Fermat’s Last Theorem; sieve of Eratosthenes

Citations:

Zbl 0588.14026