Statistical distribution of Fermat quotients

Victor Alexandru, Cristian Cobeli, Marian Vâjâitu and Alexandru Zaharescu

Chaos, Solitons & Fractals, 2022, vol. 161, issue C

Abstract: Let qpb=bp−1−1pmodp, for 0 ≤ b ≤ p2 − 1 and gcd(b,p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤp is fully covered fast by the quotients qpq with q prime and q ≪ plog2p, as p → ∞ and (ii) The matrix FQM(p) passes the pair size correlation test, in which any pair (u,v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.

Keywords: Fermat quotients; Fermat quotient matrix; Pair correlation size test; Wieferich primes (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:161:y:2022:i:c:s0960077922005458

DOI: 10.1016/j.chaos.2022.112335

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