On Krasner's Criteria for the First Case of Fermat's Last Theorem
Copyright policy )On Krasner's Criteria for the First Case of Fermat's Last Theorem
The author uses an adaptation of Krasner's method [\textit{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 \textit{Z. Šami} [Glas. Mat., III. Ser. 21(41), 259--269 (1986; Zbl 0621.10012)] who gives the same result for all \(n\) up to \([(\log p)^{2/5}]\).
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510.mathematics, p-divisibility properties, Congruences; primitive roots; residue systems, Bernoulli number, Bernoulli and Euler numbers and polynomials, Higher degree equations; Fermat's equation, Article, first case of Fermat's last theorem, numerators
510.mathematics, p-divisibility properties, Congruences; primitive roots; residue systems, Bernoulli number, Bernoulli and Euler numbers and polynomials, Higher degree equations; Fermat's equation, Article, first case of Fermat's last theorem, numerators
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