Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.3

On a Congruence of Kimball and Webb Involving Lucas Sequences


Christian Ballot
Département de Mathématiques et Mécanique
Université de Caen
F-14032 Caen Cedex
France

Abstract:

Given a pair (Ut) and (Vt) of Lucas sequences, an odd integer $\nu\ge1$, and a prime $p\ge\nu+4$ of maximal rank $\rho_U$, i.e., such that $\rho_U$ is p or $p\pm1$, we show that $\sum_{0<t<\rho_U}(V_t/U_t)^\nu \equiv0\pmod{p^2}$. This extends a result of Kimball and Webb, who proved the case $\nu=1$. Some further generalizations are also established.

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Received July 22 2013; revised versions received November 9 2013; November 28 2013. Published in Journal of Integer Sequences, December 15 2013.

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