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.
Balancing numbers were first considered by Behera and Panda in  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 , 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 , 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 , 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 number . So we set for some integer . Hence, we get the Diophantine equation: If we make the change of variables and for some and , then we get Here we obtain and . Thus (9) becomes which is a Pell equation (see [11–14]). Before considering its integer solutions, we need some notations. Now let be a nonsquare discriminant. Then the -order is defined for nonsquare discriminants to be the ring , where if mod 4, or if mod 4). So is a subring of . The unit group is defined to be the group of units of the ring .
For the quadratic form , we can write . So the module of is the -module . Therefore, we get , where So there is a bijection for solving the Pell equation ; that is, . The action of on the set of integral solutions of the equation is most interesting when is a positive nonsquare since is infinite. Therefore, the orbit of each solution will be infinite and so the set is either empty or infinite. Since can be explicitly determined, the set is satisfactorily described by the representation of such a list, called a set of representatives of the orbits. Let be the smallest unit of , that is, greater than , and let if , or if . Then every orbit of integral solutions of contains a solution such that , where if or if . So for finding a set of representatives of the orbits of integral solutions of , we must find, for each integer such that , all integers that satisfy . If , then and so .
For the Pell equation , there are one, two, three, or more sets of representatives (or sets of solutions) depending on . For example, for , the set of representatives is ; for , the set of representatives is ; for , the set of representatives is ; and for , the set of representatives is . So we cannot determine for which values of there are one, two, three, or more sets of representatives. Consequently, we cannot determine all integer solutions of in one way. To determine all integer solutions of , we have to put some restrictions on . From now on, is assumed to be odd such that is a prime.
Thus for the Pell equation , we get and is a square only for in the range . Hence, . Therefore, there is exactly one set of representatives of the orbits and that is a set of representatives; that is, there are two classes of solutions. Here we see that(1) generates the solutions for ,(2) generates the solutions for ,(3) generates the solutions for ,(4) generates the solutions for ,(5) generates the solutions for ,(6) generates the solutions for ,(7) generates the solutions for ,(8) generates the solutions for ,
where by (11). Thus, we can give the following theorem.
Theorem 1. The set of all positive integer solutions of is ,, where
To determine all positive integer solutions of , we have to formulate the th power of . So we can give the following theorem which can be proved by induction on .
Theorem 2. The th power of is where is the th balancing number and is the th Lucas-balancing number.
Consequently from the above two theorems, we can give the following main theorem.
Theorem 3. The set of all positive integer solutions of is ,, where
After formulating all integer solutions of , we can determine all integer solutions of . Notice that and . So applying Theorem 3, we get for and for . Here we note that and are positive integer solutions. By symmetry, and are also integer solutions. Apart from them, we get from generates the solutions for that for and from generates the solutions for that for . So and are also integer solutions, where and . Thus, we proved the following theorem.
Theorem 4. The set of all integer solutions of is ,, where for and for .
Example 5. Let . Then we get So the set of all integer solutions of is
After determining all integer solutions of , we can determine all -balancing numbers.
Theorem 6. The general terms of all -balancing numbers are for and for .
Proof. Since , we get from Theorem 4 that
for since and .
Note that and . So we get from (7) that Thus from (6), we obtain and hence since and .
Similarly we deduce that for .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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