On Second Order Gap Balancing Numbers

We consider the Diophantine equation for some natural numbers x, k, and r, and we call as kth order 2-gap balancing number. It was also proved that there are infinitely many first order 2-gap balancing numbers. In this paper, we show that the only second order 2-gap balancing number is 1. 1. Introduction In [1], Finkelstein defined th power numerical center for as solutions of the Diophantine equation: For , it coincides with the notion of balancing numbers introduced by Behera and Panda [2]. Finkelstein conjectured that if then there is no integer greater than with th power numerical center. Ingram in [3] proved Finkelstein's conjecture for . Further, in [4] Panda studied (1) slightly differently and called the solution of (1) as th order balancing number. The concept of gap balancing numbers was introduced by Panda and Rout [5] in connection with the Diophantine equation: They call a 2-gap balancing number or balancing number for some . Motivated by higher order balancing numbers [4] and 2-gap balancing numbers [5], we introduce higher order -gap balancing number as follows. Let be the fixed odd positive integer. We call the positive integer an th order -gap balancing number if Equation (3) is equivalent to (1) when . Similarly for fixed even positive integer , we call the positive integer an th order -gap balancing number if In this paper, we prove the following theorem. Theorem 1. The only positive integer possessing second order 2-gap balancing number is 1. 2. Background Before we prove the main result of this paper, it is better to look into the special cases corresponding to . For and (4) is equivalent to (2). Further, (2) reduces to the Diophantine equation which again reduces to the Pell's equation , ensuring infinitude of the first order 2-gap balancing numbers. Now consider the case for and arbitrary. Like the previous case (4) reduces to the Pell like equations for even and for odd which also shows infinitude of solutions of first order -gap balancing numbers. To solve (4) for and , we need the following results. Theorem 2 (see [6]). Let be nonzero integers. Then the equation has only finitely many solutions in integers . Theorem 3 (see [7]). Let be a cubic field over the field of rational numbers, and let be an integer in the ring . Suppose , where is an odd rational prime, and . Further, suppose that , where , and are rational integers, and . Then if , is never zero for any . Theorem 4 (see [8]). The Diophantine equation ( or ; if ; ; positive integers) has at most one solution in nonzero integers . There is the unique exception for the