The Diophantine Equation8x2-y2+8x(1+t)+(2t+1)2=0andt-Balancing Numbers

Merve Tayat,Meltem E. Özbek,Ahmet Tekcan

doi:10.1155/2014/897834

Abstract

Lett≥2be an odd integer such that2t2-1is a prime. In this work, we determine all integer solutions of the Diophantine equationDt:8x2-y2+8x(1+t)+(2t+1)2=0and then we deduce the general terms of allt-balancing numbers.

Highlights

Balancing numbers were first considered by Behera and Panda in [1] when they considered the integer solutions of the Diophantine equation1 + 2 + ⋅ ⋅ ⋅ + (n − 1) = (n + 1) + (n + 2) + ⋅ ⋅ ⋅ + (n + r) (1)for some positive integers n and r
The nth balancing number is denoted by Bn and the nth cobalancing number is denoted by bn
In [8], the authors generalized the theory of balancing numbers to numbers defined as follows

Summary

Introduction

The nth balancing number is denoted by Bn and the nth cobalancing number is denoted by bn. Cn is called the nth Lucas-balancing number and cn is called the nth Lucas-cobalancing number. In [8], the authors generalized the theory of balancing numbers to numbers defined as follows.

Results

Conclusion