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 8n²+1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n²+1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences B_n+1=6B_n-B_n-1 and C_n+1=6C_n-C_n-1 where B_n and C_n 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.

Received: January 11, 2013

AMS Subject Classification: 11B39, 11B83

Key Words and Phrases: balancing numbers, Lucas-balancing numbers, recurrence relation

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