Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.7

The Greatest Common Divisor of Shifted Fibonacci Numbers

Le Duc Hieu
École Polytechnique
Institut Polytechnique de Paris

Jürgen Spilker
Institute of Mathematics
University of Freiburg
79085 Freiburg im Breisgau

Luu Ba Thang
Department of Mathematics and Informatics
Hanoi National University of Education
136 Xuan Thuy
Cau Giay


Let Fn and Ln denote the Fibonacci and Lucas numbers, respectively. We generalize the well-known formula gcd(F6n+3 + 1, F6n+6 + 1) = L3n+2 if n is even. We find conditions on a, s, and t so that the functions n → gcd(Fn + s, Fn+a + s) and n → gcd(Fn + s, Fn+1 + t) are unbounded. Finally, we generalize this for three shifted Fibonacci numbers.

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(Concerned with sequences A111596 A132393.)

Received June 15 2020; revised versions received October 30 2020; January 3 2021; July 4 2021; July 22 2021; November 25 2021; January 21 2022. Published in Journal of Integer Sequences, January 26 2022.

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