The Ramanujan Journal

, Volume 14, Issue 3, pp 361–378

Estimates for Wieferich numbers

  • William D. Banks
  • Florian Luca
  • Igor E. Shparlinski

DOI: 10.1007/s11139-007-9030-z

Cite this article as:
Banks, W.D., Luca, F. & Shparlinski, I.E. Ramanujan J (2007) 14: 361. doi:10.1007/s11139-007-9030-z


We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2φ(w)≡1 (mod  w2). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above definition and demonstrate that our methods apply to such sets of integers as well.


Wieferich primes 

Mathematics Subject Classification (2000)

11A07 11N25 

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • William D. Banks
    • 1
  • Florian Luca
    • 2
  • Igor E. Shparlinski
    • 3
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMorelia, MichoacanMexico
  3. 3.Department of ComputingMacquarie UniversitySydneyAustralia

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