## 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
Full Text:

### References:

 [1] Dénes, P.: An extension of Legendre’s criterion in connection with the first case of Fermat’s last theorem. Publ. Math. Debrecen2, 115-120 (1951) · Zbl 0043.27302 [2] Elliott, P.D.T.A., Halberstam, H.: A conjecture in prime number theory. Symposia Mathematica,4, 59-72 (Instituto Nazionale di Alta Matematica, Roma, 1968/1969) [3] Fouvry, E.: Théorème de Brun-Titchmarsh. Application au Théorème de Fermat. Invent. Math.79, 383-407 (1985) · Zbl 0557.10035 [4] Furtwängler, P.: Letzter Fermatschen Satz und Eisenstein’sches Reziprozitätsgesetz. Sitzungsber. Akad. d. Wiss. Wien. Abt. II a.121, 589-592 (1912) · JFM 43.0272.02 [5] Halberstam, H., Richert, H.-E.: Sieve Methods. London: Academic Press 1974 · Zbl 0298.10026 [6] Krasner, M.: Àpropos du critère de Sophie Germain-Furtwängler pour le premier cas du théorème de Fermat. Mathematica, Cluj16, 109-114 (1940) · Zbl 0025.39402 [7] Landau, E.: Vorlesugen über Zahlentheorie. New York: Chelsea 1969 [8] Ribenboim, P.: 13 Lectures on Fermat’s Last Theorem. New York: Springer 1979 · Zbl 0456.10006 [9] Rotkiewicz, A.: Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturelsn tels quen 2|2n?2. Matematicky Vesnik.2, (17) 78-80 (1965) · Zbl 0134.27505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.