The GCD of the Shifted Fibonacci Sequence

Jürgen Spilker ()
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Jürgen Spilker: Albert-Ludwigs-Universität, Mathematisches Institut

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 473-483 from Springer

Abstract: Abstract Let f(n) be the Fibonacci-sequence defined by f(n + 2) = f(n) + f(n + 1), f(0) = 0, f(1) = 1, and let t s (n) : = gcd( f(n) + s, f(n + 1) + s), s integer. In 2011 K.-W. Chen has proved that the function t s (n) is bounded if s exceeds 1. THEOREM: Let n and s be integers. (1) t s (n) divides s 2 + (−1) n ; (2) if m := s 4 − 1 is not 0, then t s (n) is simply periodic; a period p is defined by $$f(\,p) \equiv 0\bmod m$$ , $$f(\,p + 1) \equiv 0\bmod m$$ . There are explicit formulas of t s (n) and generalisations to a wider class of recursive second-order sequences.

Keywords: Fibonacci sequence; Recurrences; Primary 11B39; Secondary 11B37 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-28203-9_28

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