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Fermat’s theorem: the contribution of Fouvry. (Théorème de Fermat: la contribution de Fouvry.) (French) Zbl 0586.10024
Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. No. 648, Astérisque 133/134, 309-318 (1986).
This lecture describes the work of Adleman, Heath-Brown and Fouvry, in showing that the first case of Fermat’s Last Theorem holds for infinitely many primes [L. M. Adleman and D. R. Heath-Brown, Invent. Math. 79, 409–416 (1985; Zbl 0557.10034); É. Fouvry, ibid. 79, 383–407 (1985; Zbl 0557.10035)]. That is to say there are infinitely many prime $$p$$ such that $$x^p+y^p=z^p$$ implies $$p\mid xyz$$. The work of Adleman and Heath-Brown reduces the problem to the demonstration of the hypothesis
$\#\{p\leq x: p\equiv 2\pmod 3,\;P(p-1)\geq x^{\vartheta}\}\gg x/\log x \tag{*}$
for some $$\vartheta >2/3$$ (where $$P(n)$$ denotes the greatest prime factor of $$n$$). Fouvry’s contribution is the proof of this hypothesis.
The lecture concentrates on the estimate (*) and describes the application of the Brun-Titchmarsh theorem, the Bombieri-Vinogradov Theorem, the “almost-all” version of the Brun-Titchmarsh theorem, the Rosser sieve with Iwaniec’s bilinear form for the remainder sum, Weil’s bound for Kloosterman sums, and the parity phenomenon. Two appendices present the Adleman-Heath-Brown criterion, and the connection between Kloosterman sums and modular forms.
For the entire collection see [Zbl 0577.00004].
##### MSC:
 11N35 Sieves 11D41 Higher degree equations; Fermat’s equation 11L05 Gauss and Kloosterman sums; generalizations
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