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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 \([(\log p)^{2/5}]\).

MSC:

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

[1] W. KELLER and G. LÖH, The Criteria of Kummer and Mirimanoff extended to include 22 Consecutive Irregular Pairs, Tokyo J. Math., 6, (1983), 397-402 · Zbl 0533.10009
[2] M. KRASNER, Sur le premier cas du théorème de Fermat, C. R. Acad. Sci. Paris, 199, (1934), 256-258 · Zbl 0010.00702
[3] D. H. LEHMER, On Fermat’s quotient, base two, Math. Comp., 36 (1981), 289-290 · Zbl 0452.10001
[4] P. RIBENBOIM, 13 Lectures on Fermat’s Last Theorem, Springer, New York-Heidelberg-Berlin, 1979 · Zbl 0456.10006
[5] Z. SAMI, On the first case of Fermat’s Last Theorem, to appear in Glasnik Matematicki, 21 (1986)
[6] S. S. WAGSTAFF, Jr., The irregular primes to 125,000, Math. Comp., 32, (1978), 583-591 · Zbl 0377.10002
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