Fermat Quotients for Composite Moduli

A search for Wieferich and Wilson primes

“…Analysis and search is underway for such composites [1]. We were able to offer aid to that project, by using the factorial sieve and arithmetic progression based algorithms of §2 to calculate P (N, N ) rapidly, finding the new instances N = 558771, 1964215, 8121909 and 12326713.…”

Wilson quotients for composite moduli

“…We call such numbers "Wilson numbers". The problem is similar to that concerning the Wieferich numbers (see [2]), but it appears to be more difficult. While the Wieferich numbers have been completely characterized in [2], no such characterization was found for the composite Wilson numbers.…”

“…The problem is similar to that concerning the Wieferich numbers (see [2]), but it appears to be more difficult. While the Wieferich numbers have been completely characterized in [2], no such characterization was found for the composite Wilson numbers. Moreover, Kloss [8] The main purpose of this section is to derive a number of congruences for Wilson quotients, some of which will facilitate the search for further composite Wilson numbers.…”

“…Note that by Euler's Theorem, this quotient is an integer. It was first studied by Lerch [9]; for further properties, see [2]. The two quotients just defined are related by the following fundamental congruence.…”

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.…”