2020/04/13 by Van Dinh, Ngô
#11B37 #11B83 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2004.05949
Balancing numbers n are originally defined as the solution of the Diophantine equation 1+2+⋯+(n-1)=(n+1)+⋯+(n+r), where r is called the balancer corresponding to the balancing number n. By slightly modifying, n is the cobalancing number with the cobalancer r if 1+2+⋯+n=(n+1)+⋯+(n+r). Let Bn denote the nth balancing number and bn denote the nth cobalancing number. Then 8Bn2+1 and 8bn2+8bn+1 are perfect squares. The nth Lucas-balancing number Cn and the nth Lucas-cobalancing number cn are the positive roots of 8Bn2+1 and 8bn2+8bn+1, respectively. In this paper, we establish some trigonometric-type identities and some arithmetic properties concerning the parity of balancing, cobalancing, Lucas-balancing and Lucas-cobalancing numbers.