The p -Adic Valuation of Lucas Sequences

2016/05/01 by Carlo Sanna · 21 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Combinatorics #Economics #Fibonacci number #Fibonacci polynomials #Lucas number #Lucas sequence #Mathematical Dynamics and Fractals #Mathematical economics #Mathematics #Valuation (finance) #advanced mathematical theories

paper · doi:10.1080/00150517.2016.12427821

published in The Fibonacci Quarterly 54(2), 118-124 (The Fibonacci Association)

openalex publication_date 2016/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

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”.

Citations

• Theorie des Fonctions Numeriques Simplement Periodiques
• The Order of the Fibonacci and Lucas Numbers

Cited by

• The p -adic Valuations of Möbius Duals of Lucas Sequences
• The p-adic Valuations of Möbius Duals of Lucas Sequences
• Numerical semigroups from rational matrices III: semigroups of matricial dimension two and a counterexample to the lonely element conjecture
• On the p-adic valuation of stirling numbers of the first kind

Related

• The p -adic Valuation of Lucas Sequences When p Is a Special Prime
• On alternative definition of Lucas atoms and their p-adic valuations
• Cyclotomic Congruences and Lucas Sequences
• The p -adic Valuations of Möbius Duals of Lucas Sequences
• The p-adic Valuations of Möbius Duals of Lucas Sequences