On vanishing Fermat quotients and a bound of the Ihara sum

2011/04/20 by Igor E. Shparlinski, Shparlinski, Igor E. · 2 citations
Mathematics · Computer Science · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1104.3910

Abstract

We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients q_p(ℓ) modulo a prime p when ℓ runs through primes ℓ ≤ N. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields \Q(exp(2 πi/p²)).

Cited by

• Intersections of multiplicative subgroups and Heilbronn's exponential sum
• Pattern formation Statistics on Fermat Quotients

Related

• Some works of Furtwängler and Vandiver revisited and Fermat's last theorem
• Double exponential lower bounds for possible solutions in the Second Case of the Fermat Last Theorem
• On Fermat's Last Theorem over the ℤ₃-extension of ℚ and other fields
• Remarques sur le premier cas du théorème de Fermat sur les corps de nombres
• On the asymptotic Fermat's Last Theorem over number fields