On the divisibility of Fermat quotients

Abstract

We show that for a prime p the smallest a with a p−1 ≡ 1 (mod p 2 ) 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) .

Artin's Primitive Root Conjecture – A Survey

“…It might be true that l p ≤ 3. Bourgain et al [42] have shown that l p ≤ log 463/252+o(1) p as p tends to infinity. As a consequence, they derive a stronger version of Lenstra's squarefree test.…”

On Heilbronn's Exponential Sum

“…Recently, some progress in estimating of exponential sums over "large" subgroups (but in Z/pZ not in Z/p 2 Z) such as (2) was attained (see [10]). So it is natural to try to use the approach from the paper to obtain some new upper bound for (1). Unfortunately, the methods from [10] cannot be applied directly in the case.…”

“…Then |S(a)| ≪ p Heilbronn's exponential sum is connected (see e.g. [1], [2], [5], [6], [11], [12]) with so-called Fermat quotients defined as q(n) = n p−1 − 1 p , n = 0 (mod p) .…”

“…By l p denote the smallest n such that q(n) = 0 (mod p). In [1] an upper bound for l p was obtained.…”

Open Problems on Exponential and Character Sums

“…It is quite possible that a combination of [74,Proposition 3] with the results about nonvanishing Fermat quotient of [31,128] may lead to a result with a rather smooth s. Problem 63. Design and efficient algorithm, that for given x and y generates a "random" y-smooth number n ≤ x, that is, the output is always y-smooth and any y-smooth number appears with probability close to 1/ψ(x, y) where as usual ψ(x, y) = #{s ≤ x : s is y-smooth}.…”