Étude probabiliste des quotients de Fermat

Abstract

For fixed $a\geq 2$, we suggest that the probability of nullity mod $p$ of the Fermat quotient $q_p(a)$ is $\ll \frac{1}{p}$ for $p \to \infty$. For this we propose various heuristics (as the existence of a suitable binomial law of probability), justified by means of numerical computations and analytical results, which may imply, via the Borel--Cantelli heuristic, that $q_p(a) \ne 0$ for all $p$ except a finite number (Th. 4.9). These heuristics are based on the possible existence (with an analogous probability) of $O(\log(p))$ ``abundant'' solutions $z_i \in [2, p-1[$ which are not necessarily of the ``exceptional'' form $a^k$, $1 \leq k < \log(p)/ \log(a)$, when $q_p(a)=0$, showing the exceptional solutions as a particular case of abundant solutions, for which a law of probability is natural. We also compute the density of integers $A$ such that $q_p(A) \ne 0, \forall p \leq x$ (Th. 4.12).