Geodesic
On a congruence of Kimball and Webb involving Lucas sequences
Journal of integer sequences, Tome 17 (2014) no. 1
Given a pair $(U_{t})$ and $(V_{t})$ 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 $\sum_{0t\rho_U}(V_t/U_t)^\nu \equiv0\pmod{p^2}$. This extends a result of Kimball and Webb, who proved the case $\nu=1$. Some further generalizations are also established.
Classification : 11B39, 11A07
Keywords: Lucas sequence, rank of appearance, congruence, Wolstenholme, leudesdorf 
@article{JIS_2014__17_1_a0,
     author = {Ballot,  Christian},
     title = {On a congruence of {Kimball} and {Webb} involving {Lucas} sequences},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {1},
     zbl = {1342.11019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_1_a0/}
}
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Ballot,  Christian. On a congruence of Kimball and Webb involving Lucas sequences. Journal of integer sequences, Tome 17 (2014) no. 1. http://geodesic.mathdoc.fr/item/JIS_2014__17_1_a0/