On the 𝑝-divisibility of the Fermat quotients

Johnson, Wells

doi:10.1090/S0025-5718-1978-0463091-4

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On the -divisibility of the Fermat quotients

Author: Wells Johnson
Journal: Math. Comp. 32 (1978), 297-301
MSC: Primary 10A10; Secondary 10A30
MathSciNet review: 0463091
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Abstract: Upper bounds for the power of p which divides the Fermat quotient ${q_a} = ({a^{p - 1}} - 1)/p$ are obtained, and conditions are given which imply that ${q_a}\nequiv\;0$ $\pmod p$ . The results are in terms of the number of steps in a simple algorithm which determines the semiorder of a $\pmod p$ .

[1] J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), Academic Press, London, 1971, pp. 213–222. MR 0314736 (47 #3288)
[2] Wells Johnson, On the nonvanishing of Fermat quotients (𝑚𝑜𝑑 𝑝), J. Reine Angew. Math. 292 (1977), 196–200. MR 0450193 (56 #8489)
[3] W. MEISSNER, "Uber die Lösungen der Kongruenz ${x^{p - 1}} \equiv 1\;\bmod \,{p^m}$ und ihre Verwertung zur Periodenbestimmung $\bmod \,{p^x}$ ," Sitzungsber. Berlin Math. Gesell., v. 13, 1914, pp. 96-107.
[4] D. MIRIMANOFF, Comptes Rendus Paris, v. 150, 1910, pp. 204-206.
[5] M. Perisastri, On Fermat’s last theorem. II, J. Reine Angew. Math. 265 (1974), 142–144. MR 0337762 (49 #2531)

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