Abstract:

Given a pair (U_t) and (V_t) of Lucas sequences, Kimball and Webb showed that $\sum_{0<t<\rho_U}\frac{V_t}{U_t} \equiv 0$ (mod p²), if p is a prime $\ge5$ whose rank $\rho_U$ is maximal, that is to say, $\rho_U$ is p or $p\pm1$. We extend their result replacing p by a composite integer m of maximal rank, thereby providing a generalization of a classical congruence of Leudesdorf.