Akademik Veri Yönetim Sistemi
Comptes Rendus de L'Academie Bulgare des Sciences, cilt.78, sa.6, ss.805-812, 2025 (SCI-Expanded, Scopus)
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 Fn + a and Fn+1 + a for any a. In 1971, it was proven that consecutive shifted Fibonacci numbers are not always relatively prime and also that gcd(Fn + a, Fn+1 + a) is unbounded for a = ±1. 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 = ±1, ±2, ±3. In this paper, the greatest common divisors are examined for consecutive Fibonacci and Lucas numbers shifted by a = ±4 and it is noticed that gcd(Fn ± 4, Fn+1 ± 4) and gcd(Ln ± 4, Ln+1 ± 4) are bounded.