The <i>p</i> -Adic Valuation of Lucas Sequences

Abstract

Let (un)n≥0 be a nondegenerate Lucas sequence with characteristic polynomial X2- aX - b, for some relatively prime integers a and b. For each prime number p and each positive integer n, we give simple formulas for the p-adic valuation νp(un), in terms of νp(n) and the rank of apparition of p in (un)n≥0. This generalizes a previous result of Lengyel on the p-adic valuation of Fibonacci numbers, and also the folkloristic “lifting-the-exponent lemma”.