Greatest common divisors of shifted balancing numbers

Abstract

It is well known that the successive balancing numbers are relatively prime. Let for all integers a, sn(a) denotes the greatest common divisor of the shifted balancing numbers of the form sn(a) = gcd(Bn 􀀀 a; Bn+1 􀀀 6a). In this study, we will show that fsn(1)g is unbounded whereas fsn(a)g is bounded for a 6= 1.

Keywords

• Balancing numbers
• Lucas-balancing numbers
• Balancing-like sequences
• Shifted balancing numbers