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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

Citations:

Zbl 0588.14026
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