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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

11N35 Sieves
11D41 Higher degree equations; Fermat’s equation
Full Text: DOI EuDML
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