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Théorème de Brun-Titchmarsh; application au théorème de Fermat. (French) Zbl 0557.10035
Denote by \(A^*_ a(x,\delta)\) the number of primes \(p\leq x\) with \(p\equiv 2 mod 3\) for which the greatest prime factor of p-a exceeds \(x^{\delta}\). It is immediate that \(A^*_ 1(x,\delta)=\sum_{x^{\delta}<p\leq x}\pi^*(x,p)\), in the notation of the preceding review. The author proves that \[ A_ a(x,\delta) \gg x/\log x\quad if\quad x>x_ 0, \] \(x_ 0\) and the implied constant depending on a and \(\delta\), for \(\delta =0.6687..\). \((>2/3)\). This is the criterion required for the application to Fermat’s last theorem reviewed above. Similar results with smaller values of \(\delta\) had been obtained by the present author [Acta Arith. 43, 417-424 (1984; Zbl 0514.10035)], J.-M. Deshouillers and H. Iwaniec [Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 319-333 (1984; Zbl 0548.10026)], and by earlier workers.
Following a method of Chebyshev as developed by C. Hooley [Mathematika 20, 135-143 (1973; Zbl 0288.10013)], where a result was obtained for any \(\delta <5/8\), it is necessary to study the quantity \(\pi (x,q,a)=| \{p; p\leq x, p\equiv a mod q\}|\) with a view to obtaining estimates of the type \(\pi (x,q,a)\leq \{C(\theta)x\}/\{\phi (q) \log x\}\) for ”almost all” q between \(x^{\theta}\) and \(2x^{\theta}\). This is then followed by an integration; the requirement is \(\int_{<\theta <\delta}C(\theta) d\theta <\). The author’s methods, which are highly technical, lead to particularly significant improvements in the available C(\(\theta)\) for values of \(\theta\) nearer to \(1/2\).
Reviewer: G.Greaves

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