Some Properties of Balancing Numbers

Abstract

In this paper we discuss some aspects and properties of Balancing Numbers and some other related numbers. We prove, among other things, that a balancing number cannot be a power of a prime integer. We give some identities concerning these numbers and its related numbers. We use linear algebra techniques to write a balancing number and its related numbers in the Binet form.

Key words and phrases. Balancing numbers, balancers, balancing matrices, Binet form, Diophantine equation, Lucas-balancing numbers, Pell’s equation.

2020 Mathematics Subject Classification. 11A05; 11B37; 11B39; 11B83