On the -divisibility of the Fermat quotients

Author:
Wells Johnson

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Upper bounds for the power of *p* which divides the Fermat quotient are obtained, and conditions are given which imply that . The results are in terms of the number of steps in a simple algorithm which determines the semiorder of *a* .

**[1]**J. BRILLHART, J. TONASCIA & P. WEINBERGER, "On the Fermat quotient,"*Computers in Number Theory*(A. O. L. Atkin & B. J. Birch, Editors), Academic Press, New York, 1971, pp. 213-222. MR**47**#3288. MR**0314736 (47:3288)****[2]**W. JOHNSON, "On the nonvanishing of Fermat quotients ,"*J. Reine Angew. Math.*, v. 292, 1977, pp. 196-200. MR**0450193 (56:8489)****[3]**W. MEISSNER, "Uber die Lösungen der Kongruenz und ihre Verwertung zur Periodenbestimmung ," 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.*, v. 265, 1974, pp. 142-144. MR**49**#2531. MR**0337762 (49:2531)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0463091-4

Keywords:
Fermat quotients,
Fermat's Last Theorem,
Wieferich's criterion

Article copyright:
© Copyright 1978
American Mathematical Society