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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])
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Abstract

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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

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