Greatest Common Divisors in Shifted Fibonacci Sequences

Kwang-Wu Chen

Department of Mathematics and Computer Science Education

Taipei Municipal University of Education No. 1, Ai-Kuo West Road Taipei, Taiwan 100, R.O.C.

                  kwchen@tmue.edu.tw

Abstract

It is well known that successive members of the Fibonacci sequence are relatively prime. Let

f_{n}(a) = gcd(F_{n} + a, F_{n+1} + a).

Therefore (f_{n}(0)) is the constant sequence 1, 1, 1, …, but Hoggatt in 1971 noted that (f_{n}(±1)) is unbounded. In this note we prove that (f_{n}(a)) is bounded if a ≠ ±1.