On perfect powers that are sum of two balancing numbers

TL;DR

This paper completely characterizes solutions to equations involving sums and differences of balancing numbers raised to powers, revealing new insights into their algebraic structure and solutions.

Contribution

It provides the first comprehensive solution set for Diophantine equations involving balancing numbers and perfect powers, under specific modular and coprimality conditions.

Findings

All solutions to B_n + B_m = x^q with n ≡ m mod 2 are identified.

Solutions to B_n^3 ± B_m^3 = x^q with q ≥ 3 and gcd(B_n, B_m) = 1 are characterized.

New bounds and properties of balancing numbers related to perfect powers are established.

Abstract

Let $B_k$ denote the $k^{th}$ term of balancing sequence. In this paper we find all positive integer solutions of the Diophantine equation $B_n+B_m = x^q$ in variables $(m, n,x,q)$ under the assumption $n\equiv m \pmod 2$. Furthermore, we study the Diophantine equation \[B_n^{3}\pm B_m^{3} = x^q\] with positive integer $q\geq 3$ and $\gcd(B_n, B_m) =1$.