Balancing numbers as sum of same power of consecutive balancing numbers (Q6148896)

The sequence of balancing numbers \( \{B_n\}_{n\ge 0} \) is defined by the binary recurrence \( B_0=0 \), \( B_1=1 \), and \( B_{n+!}=6B_n-B_{n-1} \) for all \( n\ge 1 \). In the paper under review, the authors study the Diophantine equation \[ B_n^{x}+B_{n+1}^{x}+\cdots +B_{n+k-1}^{x}=B_m,\tag{1} \] in positive integers \( (m,n,k,x) \). Their main result is the following. Theorem 1. The Diophantine equation (1) has only the trivial solutions given by \( (m,n,k,x)\in \{(n,n,1,1), (1,1,1,x)\} \). The proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the usual properties of the balancing sequence, Baker's theory for nonzero lower bounds for linear forms in logarithms of algebraic numbers, and the reduction techniques involving the theory of continued fractions. All computations are done with the aid of a computer program in \texttt{Maple}.