On Krasner’s criteria for the first case of Fermat’s last theorem. (English) Zbl 0599.10013

The author uses an adaptation of Krasner’s method [M. Krasner, C. R. Acad. Sci., Paris 199, 256–258 (1934; Zbl 0010.00702)] to prove that if the first case of Fermat’s last theorem is false for the prime \(p\), then \(p\) divides the numerator of the Bernoulli number \(B_{p-1-n}\) for all \(n\) between \(1\) and \([\sqrt{\log p/\log \log p}]\). This improves a result of Z. Šami [Glas. Mat., III. Ser. 21(41), 259–269 (1986; Zbl 0621.10012)] who gives the same result for all \(n\) up to \([(\log p)^{2/5}]\).


11D41 Higher degree equations; Fermat’s equation
11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
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