NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCAS-BALANCING NUMBERS

Abstract

Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation 1 + 2 + ··· + (n − 1) = (n + 1) + (n + 2) + ··· + (n + r). If n is a balancing number, then 8n 2 + 1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n 2 + 1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences Bn+1 = 6Bn − Bn 1 and Cn+1 = 6Cn − Cn 1 where Bn and Cn are respectively denoted by the n th balancing number and n th Lucas-balancing number. In this study, we establish some new identities for the common factors of both balancing and Lucas-balancing numbers. AMS Subject Classification: 11B39, 11B83