caa a22 4500 463177991 CHVBK 20180406164830.0 cr unu---uuuuu 170326e20071201xx s 000 0 eng 10.1007/s11139-007-9030-z doi (NATIONALLICENCE)springer-10.1007/s11139-007-9030-z Estimates for Wieferich numbers [Elektronische Daten] [William Banks, Florian Luca, Igor Shparlinski] 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. Springer Science+Business Media, LLC, 2007 Wieferich primes nationallicence Banks William Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA aut Luca Florian Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacan, Mexico aut Shparlinski Igor Department of Computing, Macquarie University, 2109, Sydney, NSW, Australia aut The Ramanujan Journal Springer US; http://www.springer-ny.com 14/3(2007-12-01), 361-378 1382-4090 14:3<361 2007 14 11139 https://doi.org/10.1007/s11139-007-9030-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-007-9030-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Banks William Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA aut NATIONALLICENCE 700 1- Luca Florian Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacan, Mexico aut NATIONALLICENCE 700 1- Shparlinski Igor Department of Computing, Macquarie University, 2109, Sydney, NSW, Australia aut NATIONALLICENCE 773 0- The Ramanujan Journal Springer US; http://www.springer-ny.com 14/3(2007-12-01), 361-378 1382-4090 14:3<361 2007 14 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer