Some algebraic relations on balancing numbers

Abstract

In this work we deduce some new algebraic relations on balancing numbers and their relationships with Pell, Pell-Lucas and oblong numbers. Also we formulate the sums of balancing, Pell and Pell-Lucas numbers in terms of balancing numbers. We deduce some relations on perfect squares, Pythagorean triples, congruent numbers, circulant matrices and spectral norms involving balancing numbers. Later, we formulate the integer solutions of some specific Pell equations via terms of balancing numbers.