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

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

Zbl 0588.14026
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