On the 𝑝-divisibility of Fermat quotients

Abstract

The authors carried out a numerical search for Fermat quotients Q a = ( a p − 1 − 1 ) / p Q_{a} = (a^{p-1}-1)/p vanishing mod p p , for 1 ≤ a ≤ p − 1 1 \le a \le p-1 , up to p > 10 6 p > 10^{6} . This article reports on the results and surveys the associated theoretical properties of Q a Q_{a} . The approach of fixing the prime p p rather than the base a a leads to some aspects of the theory apparently not published before.