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On Krasnér’s theorem for the first case of Fermat’s last theorem. (English) Zbl 0813.11015

The solvability of the Fermat equation \(x^ p + y^ p = z^ p\) \((p\) an odd prime) in the so-called first case would imply that \(p\) divides the Bernoulli numbers \(B_{p-n-1}\) for all \(n\) up to \(k(p) = 2[(\log p)^{1/3}]\). This is the content of M. Krasner’s classical theorem supplemented by a computational result. The author studies various ways of increasing the bound \(k(p)\). By modifying an argument in Krasner’s proof he achieves the bound \(\max \{(2(\log p)/ \log \log p)^{1/2}\), \((\log p)^{617/1398}\}\), which is slightly better than a bound obtained by A. Granville [Manuscr. Math. 56, 67-70 (1986; Zbl 0599.10013)].

MSC:

11D41 Higher degree equations; Fermat’s equation
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 0599.10013
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