The terms in Lucas sequences divisible by their indices

2009/08/26 by Christopher Smyth, Smyth, Chris
Mathematics · Physics and Astronomy · #11B39 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0908.3832

openalex publication_date 2009/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(aⁿ-bⁿ)/(a-b), v_n=aⁿ+bⁿ, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for which n divides u_n and also the set of indices n for which n divides v_n. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called basic number, which can only be 1, 6 or 12, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.

Related

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• On the divisibility of the rank of appearance of a Lucas sequence
• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• On a divisibility relation for Lucas sequences
• Integers represented by Lucas sequences