Research Article | Open Access
Juanli Su, Xiaoxue Li, "The Exponential Diophantine Equation ", Abstract and Applied Analysis, vol. 2014, Article ID 670175, 5 pages, 2014. https://doi.org/10.1155/2014/670175
The Exponential Diophantine Equation
Let m be a positive integer. In this paper, using some properties of exponential diophantine equations and some results on the existence of primitive divisors of Lucas numbers, we prove that if and , then the equation has only the positive integer solution .
Let , be the sets of all integers and positive integers, respectively. Given a triple of coprime positive integers with , there are many papers that investigated the equation (see [1–8]). Recently, Terai  proved that if , , satisfy then (1) has only the solution , provided that or . The proof of this result is based on elementary methods and Baker’s method. In this paper, using some properties of exponential diophantine equations and some results on the existence of primitive divisors of Lucas numbers, we prove a general result as follows.
Some Note. Combining Terai  and our Theorem, we know that only the values , are left to investigate (2). In this case, the equation has only finitely many solutions in . Moreover, they are not only effectively but also practically solvable. With some computer assistance, we would be able to solve completely the equation.
Lemma 2. If , then , and
Proof. Since , by (3), we have and . Since , we get
Since , we have and . It implies that Hence, by (3), (5), and (6), we get
If , then from (7) we get , which contradicts (5). So we have
Since is a positive integer, by (7), we have On the other hand, by (3), we have and . It implies that whence we obtain
Since , we have . Thus, by (9) and (11), we get (4). The lemma is proved.
Let , , be positive integers such that and .
Lemma 3 ([9, Lemmas 1 and 6]). If is a fixed solution of the equation then there exists a unique positive integer such that The positive integer is called the characteristic number of the solution and is denoted by the symbol .
Lemma 4 ([9, Theorems 1 and 3]). For a fixed characteristic number , let be the set of all solutions of (13) with . Then one has the following. (i) has a unique solution satisfying , and , where through all solutions of . Such is called the least solution of . (ii)Every element of can be expressed as
Lemma 5. If and , then the equality cannot hold.
Proof. If (15) holds, then
By (15) and (17), we have
We may assume that ; then, by (18), we have Since and , we see from (19) that . Further, since , by (16), we get It implies that .
Let be an odd prime divisor of . Since , we have . Further let Since , by (21), we have Further, by (21) and (22), we get Therefore, by (16), (21), and (23), we obtain Putting through all odd prime divisors of , we get from (24) that and By (19) and (25), we have and But (26) is impossible for any positive integer . Thus, (15) is false. The lemma is proved.
Let be algebraic integers. If and are nonzero coprime integers and is not a root of unity, then is called a Lucas pair. Further, let and . Then we have where . We call the parameters of the Lucas pair . Two Lucas pairs and are equivalent if . Given a Lucas pair , one defines the corresponding sequence of Lucas numbers by For equivalent Lucas pairs and , we have for any . A prime is called a primitive divisor of if and . A Lucas pair such that has no primitive divisors will be called an -defective Lucas pair. Further, a positive integer is called totally nondefective if no Lucas pair is -defective.
Lemma 6 (see ). Let satisfy and . Then, up to equivalence, all parameters of n-defective Lucas pair are given as follows:(i), , , , , , , ,(ii), ,(iii), ,(iv), ,(v), ,(vi), .
Lemma 7 (see ). If , then is totally nondefective.
Let be positive integers such that and .
Lemma 8 (see [9, Theorems 1 and 3]). Every solution of the equation
can be expressed as
where are positive integers satisfying
where is the class number of positive binary quadratic primitive forms of discriminant .
For any positive integer , let denote the set of distinct prime divisors of .
Proof. Let be a solution of (29) with . By Lemma 8, , and satisfy (30) and (31), where , and satisfy (32). Let
By (32) and (33), we have that , , , and satisfies . It implies that is a Lucas pair with parameters . Let denote the corresponding Lucas numbers. By (28), (31), and (33), we get
Since , by the definition of primitive divisors, we see from (34) that either or and the Lucas number has no primitive divisor.
If , then from (30) and (32) we get . If , by Lemmas 6 and 7, using an easy computation, the solution satisfies the case (i) or (ii). Thus, the lemma is proved.
Lemma 10 ([12, Theorems and ]). For any positive integer , one has
3. Proof of Theorem
In addition, (37) has another solution Let . We have By (39) and (41), we get . It implies that the solutions (38) and (40) belong to a same class of solutions of (37). Further, since , (40) is the least solution of . Therefore, applying Lemma 4 to (38), we get , , and However, since and , we have and . By Lemma 5, (42) is false.
We finally consider the case that . Then the equation has a solution Since , applying Lemma 8 to (44), we have where , are positive integers satisfying If , then from (46) we get where are integers satisfying By (49), we have Further, since , we see from (50) that , and Furthermore, by (51), we get ; whence we obtain . But, since , it is impossible. So we have .
Notice that the solution (44) satisfies . Therefore, since and , by Lemma 9, we have and Further, applying Lemma 10 to (56), we get On the other hand, by Lemma 2, satisfies (4). The combination of (4) and (57) yields But (58) is false for . Thus, the solution with does not exist. The theorem is proved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11071194) of China, the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 12JK0871), and the Scientific Research Program Funded by Yangling Vocational and Technical College (Program no. A2013027).
- Y. Fujita and T. Miyazaki, “Jeśmanowicz' conjecture with congruence relations,” Colloquium Mathematicum, vol. 128, no. 2, pp. 211–222, 2012.
- B. He, A. Togbé, and S.-C. Yang, “On the solutions of the exponential Diophantine equation ,” Quaestiones Mathematicae, vol. 35, no. 1, pp. 1–17, 2012.
- F. Luca, “On the system of Diophantine equations and ,” Acta Arithmetica, vol. 153, no. 4, pp. 373–392, 2012.
- T. Miyazaki, “Terai's conjecture on exponential Diophantine equations,” International Journal of Number Theory, vol. 7, no. 4, pp. 981–999, 2011.
- T. Miyazaki, “Jeśmanowicz' conjecture on exponential Diophantine equations,” Functiones et Approximatio Commentarii Mathematici, vol. 45, no. 2, pp. 207–229, 2011.
- T. Miyazaki, “Generalizations of classical results on Jeśmanowicz' conjecture concerning Pythagorean triples,” Journal of Number Theory, vol. 133, no. 2, pp. 583–595, 2013.
- T. Miyazaki and A. Togbé, “The Diophantine equation ,” International Journal of Number Theory, vol. 8, no. 8, pp. 2035–2044, 2012.
- N. Terai, “On the exponential Diophantine equation ,” International Journal of Algebra, vol. 6, no. 21–24, pp. 1135–1146, 2012.
- M. Le, “Some exponential Diophantine equations. I. The equation ,” Journal of Number Theory, vol. 55, no. 2, pp. 209–221, 1995.
- P. M. Voutier, “Primitive divisors of Lucas and Lehmer sequences,” Mathematics of Computation, vol. 64, no. 210, pp. 869–888, 1995.
- Yu. Bilu, G. Hanrot, and P. M. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers,” Journal für die Reine und Angewandte Mathematik, vol. 539, pp. 75–122, 2001.
- L. K. Hua, Introduction to Number Theory, Springer, Berlin, Germany, 1982.
Copyright © 2014 Juanli Su and Xiaoxue Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.