Estimates for Wieferich numbers

William D. Banks*, Florian Luca, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We define Wieferich numbers to be those odd integers w ≥3 that satisfy the congruence 2 φ(w) ≡1 (mod∈∈w 2). 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.

Original languageEnglish
Pages (from-to)361-378
Number of pages18
JournalRamanujan Journal
Issue number3
Publication statusPublished - Dec 2007


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