On the Divisibility of Fermat Quotients
نویسندگان
چکیده
We show that for a prime p the smallest a with ap−1 6≡ 1 (mod p2) does not exceed (log p)463/252+o(1) which improves the previous bound O((log p)2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p)5/3+o(1).
منابع مشابه
On the p-divisibility of Fermat quotients
The authors carried out a numerical search for Fermat quotients Qa = (ap−1 − 1)/p vanishing mod p, for 1 ≤ a ≤ p − 1, up to p < 106. This article reports on the results and surveys the associated theoretical properties of Qa. 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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