Etude probabiliste des p-quotients de Fermat

TL;DR

This paper investigates the probability that the p-Fermat quotient q(p,a) equals zero, suggesting it is much lower than 1/p, supported by heuristics, numerical data, and analytical insights, but without definitive proofs.

Contribution

It introduces new heuristics and analytical approaches to estimate the probability of nullity of p-Fermat quotients, proposing potential finiteness and existence results.

Findings

Heuristics indicate low probability of q(p,a)=0 for large p

Numerical computations support the heuristics

Analytical results suggest possible finiteness of solutions

Abstract

For a fixed integer a>1, we suggest that the probability of nullity of the p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime number p. For this we use various heuristics, justified by means of numerical computations and analytical results, which may imply the finiteness of the q(p,a) equal to 0 and the existence of integers a such that q(p,a) is different from 0 for all p. However no proofs are obtained concerning these heuristics.