Fibonacci numbers which are products of two balancing numbers

Abstract

The Fibonacci sequence (Fn) is defined by F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 2. The balancing number sequence (Bn) is defined by B0 = 0, B1 = 1 and Bn = 6Bn−1 − Bn−2 for n ≥ 2. In this paper, we find all Fibonacci numbers which are products of two balancing numbers. Also we found all balancing numbers which are products of two Fibonacci numbers. More generally, taking k,m,m as positive integers, it is proved that Fk = BmBn implies that (k,m, n) = (1, 1, 1), (2, 1, 1) and Bk = FmFn implies that (k,m, n) = (1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 3, 4).