Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T02:35:31.684Z Has data issue: false hasContentIssue false
  • English
  • Français

Repunit Lehmer numbers

Part of: Sequences and sets Multiplicative number theory

Published online by Cambridge University Press:  28 October 2010

Javier Cilleruelo  and
Florian Luca
Javier Cilleruelo
Affiliation:
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain ([email protected])
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, CP 58089, Morelia, Michoacán, México ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

Keywords

Lehmer numbersrepunitsprimitive divisors

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 55 - 65
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1.Bang, A. S., Taltheoretiske Undersøgelser, Tidsskrift Math. 5 (1886), 70–80, 130–137.Google Scholar
2.Cohen, G. L. and Hagis, P., On the number of prime factors of n if ϕ(n)|n − 1, Nieuw Arch. Wisk. 28 (1980), 177–185.Google Scholar
3.Guy, R. K., Unsolved problems in number theory (Springer, 2004).Google Scholar
4.Lehmer, D. H., On Euler's totient function, Bull. Am. Math. Soc. 38 (1932), 745–751.Google Scholar
5.Luca, F., Fibonacci numbers with the Lehmer property, Bull. Polish Acad. Sci. Math. 55 (2007), 7–15.Google Scholar
6.Pomerance, C., On composite n for which ϕ(n)|n – 1, II, Pac. J. Math. 69 (1977), 177–186.Google Scholar