On perfect powers that are sum of two balancing numbers

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 $n\equiv m(\mathrm{mod}2)$. Furthermore, we study the Diophantine equation \[B_n^{3}\pm B_m^{3} = x^q\] with positive integer $q\ge 3$ and $\gcd (B_n,B_m)=1$.

10 pages

References in corpus (1)

• On the resolution of the Diophantine equation $U_n+U_m=x^q$