On the Diophantine equation \(B_{n_{1}} + B_{n_{2}} = 2^{a_{1}} + 2^{a_{2}} + 2^{a_{3}}\) (Q6585129)

In the paper under review, the authors consider the Diophantine equation \N\[\NB_{n_1}+B_{n_2}=2^{a_1}+2^{a_2}+2^{a_3}, \tag{1}\N\]\Nin non-negative integers \( (n_1,n_2,a_1,a_2,a_3) \), where \( B_n \) denotes the \( n \)-th balancing number that satisfies the binary recurrence relation \( B_0=0\), \( B_1=1 \), and \( B_{n+1}=6B_n-B_{n-1} \) for all \( n\ge 1 \).\N\NTheir main result is the following. \N\NTheorem 1. All non-negative integer solutions \( (n_1,n_2,a_1,a_2,a_3) \) of the Diophantine equation (1) are given by\N\begin{align*}\N(n_1,n_2,a_1,a_2,a_3) \in \{ &(2,0,1,1,1,), (2,0,2,0,0), (2,1,2,1,0), (2,2,2,2,2), (2,2,3,1,1),\\\N& (3,0,5,1,0), (3,1,4,4,2), (3,1,5,1,1), 3,2,5,3,0), (3,3,6,2,1)\}.\N\end{align*}\NThe 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, as well as reduction techniques involving the theory of continued fractions. All numerical computations are done with the aid of a simple computer program in \texttt{Mathematica}.