On the 𝑝-divisibility of Fermat quotients

Abstract

Abstract. The authors carried out a numerical search for Fermat quotients Qa = (a p−1 − 1)/p vanishing mod p, for 1 ≤ a ≤ p − 1, up to p < 10 6 . 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.

On the divisibility of Fermat quotients

“…Congruences with Fermat quotients q p (a) modulo higher powers of p have also been considered in the literature, see [5,12]. Using our approach with bounds of generalized Heilbronn sums for fixed m ≥ 2.…”

“…It is well known that divisibility of Fermat quotients q p (a) by p has numerous applications which include the Fermat Last Theorem and squarefreeness testing, see [5,6,7,15]. In particular, the smallest value p of a for which q p (a) ≡ 0 (mod p) plays a prominent role in these applications.…”

Structure of Pseudorandom Numbers Derived from Fermat Quotients

“…There are several results which involve the distribution and structure of Fermat quotients q p (u) modulo p and it has numerous applications in computational and algebraic number theory, see e.g. [8,10,11,21] and references therein.…”

Congruences involving the Fermat quotient

“…This quotient has been extensively studied because of its links to numerous question in number theory. It is well known that divisibility of the Fermat quotient q p (a) by p has numerous applications which include the Fermat Last Theorem and squarefreeness testing (see [1], [4], [6], [12], [16], [22], [27] and [30]). In particular, solvability of the congruence q p (2) ≡ 0 (mod p) for a prime p with p ≡ 1 (mod 4) and the congruences q p (a) ≡ 0 (mod p) with a ∈ {2, 3, 5} were studied by S. Jakubec in [18] and [19], respectively.…”