An exponential Diophantine equation related to the difference between powers of two consecutive Balancing numbers

Abstract

In this paper, we find all solutions of the exponential Diophantine equation +1 − = in positive integer variables ( , , ), where is the -th term of the Balancing sequence.

On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

“…(3) has no positive integer solution (k, n, m, x) with k ≥ 3, n ≥ 2, and x ≥ 1. Another related result involving the balancing numbers was studied by Rihane et al in [10].…”

“…The same applies to V. All the above formulas given in Eqs. (8), (9), (10), (11), and (13) hold when the indices are arbitrary integers, not necessarily nonnegative.…”

On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

“…( 3) has no positive integer solution (k, n, m, x) with k ≥ 3, n ≥ 2, and x ≥ 1. Another related result involving the balancing numbers was studied by Rihane et al in [17].…”

An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers

“…If Λ 2 = 0, then we see that α 2(nx−r) = 5 x , which is possible only when nx = r since 5 x ∈ Z. This is impossible since r < nx by (14). Therefore Λ 2 = 0.…”

“…In [13], Rihane et al tackled the Diophantine equation P x n + P x n+1 = P m and gave all the solutions of this equation in nonnegative integers m, n, x. Same authors, in [14], proved that the Diophantine equation…”