Etude probabiliste des p-quotients de Fermat

Georges Gras (1)

1 Chercheur indépendant

For fixed a>1, we suggest that the probability of nullity of the p-Fermat quotient q_p(a) is lower than 1/p for any arbitrary large prime p. For this we propose various heuristics, justified by means of numerical computations and analytical results, which may imply the finiteness of the q_p(a) equal to 0 (Theorem 4.11) and the existence of integers a such that q_p(a)≠ 0 for all p. We show that the density of integers A such that q_p(A) ≠ 0, for all p< x, is about O(1/log(x)) (Theorem 4.13).

Mots clés: Fermat quotients; cyclotomic polynomials; probabilistic number theory

Domaines: Théorie des nombres [math.NT]

Georges Gras. Etude probabiliste des p-quotients de Fermat. 2014. ⟨hal-01062305v2⟩