Diophantine equations concerning balancing and Lucas balancing numbers

In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers B n are given by the recurrence B n = 6 B n - 1 - B n - 2 with initial conditions B 0 = 0 , B 1 = 1 and its associated Lucas balancing numbers C n are given by the recurrence C n = 6 C n - 1 - C n - 2 with initial conditions C 0 = 1 , C 1 = 3 . First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution ( x , y ) for the Diophantine equation 2 x 2 + 1 = 3 b y m for any even positive integers b and m with m > 2 , given in (Int J Number Theory 11:1259–1274, 2015 ). Also we prove that the Diophantine equations B n B n + d … B n + ( k - 1 ) d = y m and C n C n + d … C n + ( k - 1 ) d = y m have no solution for any positive integers n , d , k , y , and m with m ≥ 2 , y ≥ 2 and gcd ( n , d ) = 1 .