On the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$

TL;DR

This paper completely solves a specific Diophantine equation involving balancing numbers and powers of two, identifying all positive integer solutions and advancing understanding of these special number relationships.

Contribution

It provides a complete classification of solutions to the equation involving balancing numbers and powers of two, a novel result in the study of Diophantine equations.

Findings

All solutions for the equation are explicitly determined.

The solutions involve balancing numbers and powers of two.

The result advances the understanding of additive properties of balancing numbers.

Abstract

In this study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.