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March 2016 Étude probabiliste des quotients de Fermat
Georges Gras
Funct. Approx. Comment. Math. 54(1): 115-140 (March 2016). DOI: 10.7169/facm/2016.54.1.9


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).


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Georges Gras. "Étude probabiliste des quotients de Fermat." Funct. Approx. Comment. Math. 54 (1) 115 - 140, March 2016.


Published: March 2016
First available in Project Euclid: 22 March 2016

zbMATH: 06862338
MathSciNet: MR3477738
Digital Object Identifier: 10.7169/facm/2016.54.1.9

Primary: 11K65 , 11R18
Secondary: 11N37

Keywords: Borel-Cantelli heuristic , cyclotomic polynomials , Fermat quotients , prime numbers , Probabilistic number theory

Rights: Copyright © 2016 Adam Mickiewicz University


Vol.54 • No. 1 • March 2016
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