Values of Lucas sequences modulo primes
Let \(F_n\) denote the Fibonacci numbers. It is very easy to show that primes \(p \neq 5\) divide \(F_{p \pm 1}\), but finding the correct sign involves a piece of quadratic reciprocity. \textit{M. Ward} [Pac. J. Math. 11, 379--386 (1961; Zbl 0112.26904)] has observed that the question which primes divide \(F_{(p \pm 1)/k}\) for \(k = 2, 3, 4 \ldots\) is connected with higher reciprocity laws [see also e.g. \textit{E. Lehmer}, Fibonacci Q. 4, 135--138 (1966; Zbl 0144.27405) or \textit{A. Aigner}, J. Reine Angew. Math. 274--275, 139--140 (1975; Zbl 0302.12002)]. In this article, these criteria are extended to general Lucas sequences, and in some cases the exact (nonzero) values of the terms modulo \(p\) are determined.
scientific article; zbMATH DE number 2067069