Balancing and Cobalancing Numbers

Abstract

A different approach to the theory of balancing numbers is possible by means of a Pell’s equation which can be derived from the definition of balancing numbers. Each balancing number corresponds to a balancer, and each balancer, in turn, is a cobalancing number. Similarly, each cobalancing number corresponds to a cobalancer and interestingly, each cobalancer is a balancing number. The cobalancing numbers as well as their cobalancers are also solutions of a Diophantine equation similar to that satisfied by balancing numbers and their balancers. Some Diophantine equations exhibit beautiful solutions in terms of balancing and cobalancing numbers. The Lucas-balancing and Lucas-cobalancing numbers, obtained respectively as functions of balancing and cobalancing numbers, are useful in the computation of balancing and cobalancing numbers of higher order. Pell and associated Pell numbers are very closely associated with balancing and cobalancing numbers and appear in the factorization and as greatest common divisors of these numbers. The balancing, cobalancing and other related numbers are also expressible in terms of products of matrices.