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On the $p$-divisibility of Fermat quotients

Authors: R. Ernvall and T. Metsänkylä
Journal: Math. Comp. 66 (1997), 1353-1365
MSC (1991): Primary 11A15, 11Y70; Secondary 11D41, 11R18
MathSciNet review: 1408373
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Abstract: The authors carried out a numerical search for Fermat quotients $Q_{a} = (a^{p-1}-1)/p$ vanishing mod $p$, for $1 \le a \le p-1$, up to $p < 10^{6}$. This article reports on the results and surveys the associated theoretical properties of $Q_{a}$. The approach of fixing the prime $p$ rather than the base $a$ leads to some aspects of the theory apparently not published before.

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Additional Information

R. Ernvall
Affiliation: Forssa Institute of Technology, Saksankatu 46, FIN-30100 Forssa, Finland

T. Metsänkylä
Affiliation: Department of Mathematics, University of Turku, FIN-20014 Turku, Finland

Keywords: Fermat quotients, computation, Fermat's equation, Catalan's equation, cyclotomic fields
Received by editor(s): March 4, 1996
Received by editor(s) in revised form: May 22, 1996
Article copyright: © Copyright 1997 American Mathematical Society