Powers of two as sums of two balancing numbers (Q6616845)

Let \( (B_n)_{n\ge 1} \) be the sequence of balancing numbers defined by the binary recurrence relation \( B_1=1 \), \( B_2=6 \), and \( B_n=6B_{n-1}-B_{n-2} \) for all \( n\ge 3 \). In the paper under review, the authors prove the following theorem, which is the main result in the paper.\N\NTheorem 1. The only solutions to \( B_n+B_m =2^{a}\) in positive integers \( (n,m,a) \) with \( n\ge m \) are given by \( B_1+B_1=2 \). In particular, the only balancing number which is a power of \( 2 \) is \( B_1=1 \). \N\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 non-zero lower bounds for linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions. All computations are done with the aid of a simple computer program in \texttt{Mathematica}.\N\NFor the entire collection see [Zbl 1540.05004].