The Diophantine Equation and -Balancing Numbers

Let be an odd integer such that is a prime. In this work, we determine all integer solutions of the Diophantine equation and then we deduce the general terms of all -balancing numbers. 1. Introduction Balancing numbers were first considered by Behera and Panda in [1] when they considered the integer solutions of the Diophantine equation for some positive integers and . In this case is called a balancing number with balancer (or cobalancing number) . For example, and are balancing numbers with cobalancing numbers which are and , respectively. The th balancing number is denoted by and the th cobalancing number is denoted by . They satisfy the recurrence relations and for , with initial values and . From (1), one has So is a balancing number if and only if is a perfect square and is a cobalancing number if and only if is a perfect square. Set and . Then is called the th Lucas-balancing number and is called the th Lucas-cobalancing number. Binet formulas for balancing, cobalancing, Lucas-balancing, and Lucas-cobalancing numbers are , , and , respectively, where and (for further details see also [2–7]). Recently, there are many studies on balancing numbers. In [8], the authors generalized the theory of balancing numbers to numbers defined as follows. Let such that . Then a positive integer such that is called a -power numerical center for if Later in [9], the authors extended the concept of balancing numbers to the -balancing numbers defined as follows. Let and let be coprime integers. If for some positive integers, and , one has then the number is called an -balancing number and is denoted by . Like in -balancing numbers, in [10], Dash et al. defined the -balancing number and derived some results on it. Let be an integer. Then a positive integer is called a -balancing number if for some positive integer which is called -balancer (or -cobalancing number). - and -balancing numbers can be given in terms of balancing numbers; indeed, and . So it is assumed that . The th -balancing number is denoted by and the th -cobalancing number is denoted by . From (5) we see that Hence, is a -balancing number if and only if is a perfect square and is a -cobalancing number if and only if is a perfect square. Set Then is called the th Lucas -balancing number and is called the th Lucas -cobalancing number. 2. Main Results To determine the general terms of all -balancing numbers, we have to determine all integer (in fact only positive) solutions of some specific Diophantine equations. Let us explain this as follows. We see as above that is a perfect square for a -balancing