Abstract
This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included.
Highlights
An integer n is called a balancing number if there exists another integer r, called a balancer, corresponding to n, such that the following Diophantine equation holds1 + 2 + · · · + (n − 1) = (n + 1) + (n + 2) + · · · + (n + r). (1)Citation: Chung, C.-L.; Zhong, C.; Zhou, K
The problem of determining all balancing numbers in the set of natural numbers leads to a second order linear recursive sequence or a Pell equation
The problem of determining all (a, b)-type balancing numbers in the set of arithmetic progressions leads to the equation x2 − 8y2 = c
Summary
They defined the (a, b)-type balancing numbers with the (a, b)-type balancer as solutions of the Diophantine equation (a + b) + (2a + b) + · · · + (a(n − 1) + b) = (a(n + 1) + b) + · · · + (a(n + r) + b), (3) Note that when (a, b) = (1, 0), we get nothing but the original balancing number Bm. Instead of requesting integers a > 0 and b ≥ 0, our definition of (a, b)-type balancing should only exclude from the cases a = 0 or gcd(a, b) = 1. The problem of determining all (a, b)-type balancing numbers in the set of arithmetic progressions leads to the equation x2 − 8y2 = c.
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