Keller, Wilfrid; Löh, Günter The criteria of Kummer and Mirimanoff extended to include 22 consecutive irregular pairs. (English) Zbl 0533.10009 Tokyo J. Math. 6, 397-402, Supp. 487 (1983). If the Fermat equation \(x^ p+y^ p+z^ p=0\) (\(p\) an odd prime) has a solution with \(xyz\) prime to \(p\), then \(p\) divides the Bernoulli numbers \(B_{p-3},B_{p-5},\ldots,B_{p-19}\) [see H. Wada, Tokyo J. Math. 3, 173–176 (1980; Zbl 0448.10016)]. The authors show by numerical computations that here 19 can be replaced by 45. Their method follows a program proposed by Wada [op. cit.]. The computations involve prime factorizations of certain integers having order of magnitude up to \(10^{60}\). Reviewer: Tauno Metsänkylä (Turku) Cited in 2 Documents MSC: 11D41 Higher degree equations; Fermat’s equation 11B68 Bernoulli and Euler numbers and polynomials 11-04 Software, source code, etc. for problems pertaining to number theory Keywords:Fermat last theorem; irregular prime; Bernoulli numbers; numerical computations Citations:Zbl 0448.10016 PDFBibTeX XMLCite \textit{W. Keller} and \textit{G. Löh}, Tokyo J. Math. 6, 397--402, Supp. 487 (1983; Zbl 0533.10009) Full Text: DOI