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The first case of Fermat’s last theorem. (English) Zbl 0557.10034
The ”first case” of Fermat’s last theorem states that $$x^ p+y^ p\neq z^ p$$ if the prime p satisfies $$p\nmid xyz>0$$. Denote by S the set of primes p for which this statement is true. The authors give two criteria which, they show, would imply that S is infinite. One of these criteria has been shown to hold by E. Fouvry (see the following review).
The authors establish the following statement, of which the case $$k=1$$ follows from the classical result of Sophie Germain: if $$3\nmid k$$ then the number of primes $$p\not\in S$$ for which $$2kp+1$$ is also prime is $$O(k^ 2)$$. Let $$\pi^*(x,k)$$ denote the number of primes p for which $$p\leq x$$, $$p\equiv 1 mod k$$, $$p\not\equiv 1 mod 3$$. The criterion established by Fouvry is that $\sum_{x^{\theta}<p\leq x}\pi^*(x,p)\quad \gg \quad x/\log x$ for a certain $$\theta >2/3$$. The need for the parameter 2/3 in this criterion is related to the occurrence of the bound $$O(k^ 2)$$ above. This leads to the main result, in the form $\sum_{x^{\theta}<p\leq x, p\in S}p^{-1} \log p\quad \gg \quad \log x.$ A quantitatively stronger form of this result is also shown, rather more easily, to follow from a stronger (unproved) criterion, that a theorem of Bombieri’s type holds for $$\pi^*(x,p)$$ with p as large as $$x^{\theta}$$.
Reviewer: G.Greaves

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