Some Facinating Properties of Balancing Numbers

Abstract

The study of number sequences has been a source of attraction to the mathematicians since ancient times. Since then many of them are focusing their interest on the study of the fascinating triangular numbers. In a recent study Behera and Panda tried to find the solutions of the Diophantine equation 1+2+ • • • +(n-1) = (n+1) + (n+2) + • • •+ (n+r) and found that the square of any n Îℤ+ satisfying this equation is a triangular number. It can be also shown that if r Îℤ+ satisfies the above equation then is also a triangular number. If a pair (n, r) constitutes a solution of the above equation then n is called a balancing number and r is called the balancer corresponding to n. In the joint paper “On the square roots of triangular numbers” published in “The Fibonacci Quarterly” in 1999, Behera and Panda introduced balancing numbers and studied many important properties of these numbers. In this paper we establish some other interesting arithmetic-type, de-Moivre’s-type and trigonometric-type properties of balancing numbers. We also establish a most important property concerning the greatest common divisor of two balancing numbers