## 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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