On Greatest Common Divisors of Consecutive Shifted Fibonacci Sequences

Abstract

The shifted Fibonacci numbers which are formed by adding an integer $$a$$ to the Fibonacci numbers firstly appeared in literature in 1973. It is well known that consecutive Fibonacci numbers are relatively prime. It has been a matter of curiosity among researchers that computing the greatest common divisor (gcd) of consecutive shifted Fibonacci numbers $$F_{n}+a$$ and $$F_{n+1}+a$$ for any $$a$$. In 1971, it was proven that consecutive shifted Fibonacci numbers are not always relatively prime and also that $$\gcd(F_{n}+a,F_{n+1}+a)$$ is unbounded for $$a=\pm1$$. Recently, there has been an increasing interest in the greatest common divisors of shifted Fibonacci (Lucas) numbers, and in the studies conducted, they have been investigated for $$a=\pm1,\pm2,\pm3$$. In this paper, the greatest common divisors are examined for consecutive Fibonacci and Lucas numbers shifted by $$a=\pm4$$ and it is noticed that $$\gcd(F_{n}\pm4,F_{n+1}\pm4)$$ and $$\gcd(L_{n}\pm4,L_{n+1}\pm4)$$ are bounded.