Wilson quotients for composite moduli

Abstract

Abstract. An analogue for composite moduli m ≥ 2 of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by m is investigated; such an m will be called a "Wilson number". It is shown that numbers in certain infinite classes cannot be Wilson numbers. Eight new Wilson numbers up to 500 million were found.

Fermat Quotients for Composite Moduli

“…TABLE I Number of m for Which q(a, m)#0 (mod m) ) 2 104 63 39 79 18 70 65 8 3 341 21 40 357 105 71 471 390 5 V 92 41 V 58 72 0 0 6 V 21 42 6 6 73 4 4 7 1473 144 43 56 56 74 1 1 10 688 12 44 114 78 75 851 184 11 Finally we note that the related Wilson quotients, which are based on the well-known theorem of Wilson (see, e.g., [13]) can also be extended to composite moduli, and one can study composite analogues of Wilson primes. This is the subject of a separate paper [2]. It is also worth mentioning that Fermat quotients and Fermat primes have recently been extended to function fields; see [15].…”

Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-Non-Wilson Primes 2, 3, 14771

“…Proof. Replace a with ab in equation (1). Substituting a p−1 = pq p (a) + 1 and b p−1 = pq p (b) + 1, we deduce Eisenstein's logarithmic relation [11] q p (ab) ≡ q p (a) + q p (b) (mod p) and Lerch's formula follows.…”

“…When it is, p cannot divide a, and so the Fermat quotient q p (a) is an integer. In fact, (1) shows that a prime p is a Wieferich prime base a if and only if p does not divide a but does divide q p (a).…”