Number Theory
The structure of the set of numbers with the Lehmer property
Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 727-728.

Let φ be the Euler totient function, and let k,r be fixed integers with k1 and |r|1. A positive integer n has the Lehmer property if it is composite and φ(n) divides n1. We give a short proof that the set L(k,r) – of numbers n with the Lehmer property that fulfil the extra condition φ(n)kr(modn) – is finite. This is an extension of a result obtained recently by Deaconescu.

Soit φ la fonction indicatrice d'Euler, et soient k,r des entiers fixés tels que k1 et |r|1. Un entier strictement positif n a la propriété de Lehmer s'il est composé et si φ(n) divise n1. On donne une courte preuve du fait que l'ensemble L(k,r) – des nombres n possédant la propriété de Lehmer et qui vérifient la condition supplémentaire suivante φ(n)kr(modn) – est fini. Ceci est une extension d'un résultat obtenu récemment par Deaconescu.

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Published online:
DOI: 10.1016/j.crma.2008.05.002
Wójtowicz, Marek 1; Skonieczna, Marta 1

1 Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland
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Wójtowicz, Marek; Skonieczna, Marta. The structure of the set of numbers with the Lehmer property. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 727-728. doi : 10.1016/j.crma.2008.05.002. http://www.numdam.org/articles/10.1016/j.crma.2008.05.002/

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[2] Deaconescu, M. On the equation m1=aφ(n), Integers: Electronic Journal of Combinatorial Number Theory, Volume 6 (2006) (Paper A06)

[3] Grytczuk, A.; Wójtowicz, M. On a Lehmer problem concerning Euler's totient function, Proc. Japan Acad. Ser. A, Volume 79 (2003), pp. 136-138

[4] Hagis, P. Jr. On the equation Mφ(n)=n1, Nieuw Arch. Wisk. (4), Volume 6 (1988), pp. 225-261

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[6] Luca, F. Fibonacci numbers with the Lehmer property, Bull. Pol. Acad. Sci. Math., Volume 55 (2007), pp. 7-15

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