ON GREATEST COMMON DIVISORS OF CONSECUTIVE SHIFTED FIBONACCI SEQUENCES
Comptes Rendus de L'Academie Bulgare des Sciences, vol.78, no.6, pp.805-812, 2025 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 78 Issue: 6
- Publication Date: 2025
- Doi Number: 10.7546/crabs.2025.06.02
- Journal Name: Comptes Rendus de L'Academie Bulgare des Sciences
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aquatic Science & Fisheries Abstracts (ASFA), BIOSIS, CAB Abstracts, zbMATH
- Page Numbers: pp.805-812
- Keywords: Fibonacci numbers, greatest common divisor, Lucas numbers, shifted Fibonacci sequence, shifted Lucas sequence
- Yozgat Bozok University Affiliated: Yes
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 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.