Summary: 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.

11B39, 11A07

Lucas sequence, rank of appearance, congruence, Wolstenholme, leudesdorf