On the $p$-divisibility of the Fermat quotients
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- by Wells Johnson PDF
- Math. Comp. 32 (1978), 297-301 Request permission
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 $\pmod p$.References
- 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
- Wells Johnson, On the nonvanishing of Fermat quotients $(\textrm {mod}$ $p)$, J. Reine Angew. Math. 292 (1977), 196–200. MR 450193, DOI 10.1515/crll.1977.292.196 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. D. MIRIMANOFF, Comptes Rendus Paris, v. 150, 1910, pp. 204-206.
- M. Perisastri, On Fermat’s last theorem. II, J. Reine Angew. Math. 265 (1974), 142–144. MR 337762, DOI 10.1515/crll.1974.265.142
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 297-301
- MSC: Primary 10A10; Secondary 10A30
- DOI: https://doi.org/10.1090/S0025-5718-1978-0463091-4
- MathSciNet review: 0463091