The structure of the set of numbers with the Lehmer property

Marek Wójtowicz; Marta Skonieczna

doi:10.1016/j.crma.2008.05.002

Number Theory

The structure of the set of numbers with the Lehmer property

Presented by: Jean-Pierre Serre

Marek Wójtowicz ¹; Marta Skonieczna ¹

¹ Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 727-728.

is referenced by Appendix to the Note “The structure of the set of numbers with the Lehmer property”

Abstract
Résumé

Let φ be the Euler totient function, and let $k, r$ be fixed integers with $k ⩾ 1$ and $| r | ⩾ 1$ . A positive integer n has the Lehmer property if it is composite and $φ (n)$ divides $n - 1$ . We give a short proof that the set $L (k, r)$ – of numbers n with the Lehmer property that fulfil the extra condition $φ {(n)}^{k} \equiv r (\mod n)$ – 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 $k ⩾ 1$ et $| r | ⩾ 1$ . Un entier strictement positif n a la propriété de Lehmer s'il est composé et si $φ (n)$ divise $n - 1$ . 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 $φ {(n)}^{k} \equiv r (\mod n)$ – est fini. Ceci est une extension d'un résultat obtenu récemment par Deaconescu.

Received: 2008-01-07
Accepted: 2008-04-24
Published online: 2008-06-06

DOI: 10.1016/j.crma.2008.05.002

Author's affiliations:

Marek Wójtowicz ¹; Marta Skonieczna ¹

¹ Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

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     title = {The structure of the set of numbers with the {Lehmer} property},
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Marek Wójtowicz; Marta Skonieczna. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.002/

References
Cited by

[1] G.L. Cohen; P. Hagis On the number of prime factors of n if $φ (n) | n - 1$ , Nieuw Arch. Wisk. (3), Volume 28 (1980), pp. 177-185

[2] M. Deaconescu On the equation $m - 1 = a φ (n)$ , Integers: Electronic Journal of Combinatorial Number Theory, Volume 6 (2006) (Paper A06)

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

[4] P. Hagis On the equation $M \cdot φ (n) = n - 1$ , Nieuw Arch. Wisk. (4), Volume 6 (1988), pp. 225-261

[5] D.H. Lehmer On Euler's totient function, Bull. Amer. Math. Soc., Volume 38 (1932), pp. 745-751

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

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