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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 186416, 4 pages

http://dx.doi.org/10.1155/2014/186416
Research Article

On the Odd Prime Solutions of the Diophantine Equation

School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Received 16 May 2014; Accepted 7 July 2014; Published 13 July 2014

Academic Editor: Jinde Cao

Copyright © 2014 Yuanyuan Deng and Wenpeng Zhang. 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.

Abstract

Using the elementary method and some properties of the least solution of Pell’s equation, we prove that the equation has no positive integer solutions ( ) with and being odd primes.

1. Introduction

Let , be the sets of all integers and positive integers, respectively. In recent years, there are many authors who investigated the various properties of exponential diophantine equation with circulating form (see [14]). Recently, Zhang et al. [5] are interested in the equation Using the -adic lower bound of the log-linear model method, they proved all solutions of (1) satisfying . Meanwhile, they proposed a conjecture as follows.

Conjecture 1. Equation (1) has no positive integer solution .

Using the method in [5], it seems to be a very difficult problem to improve the upper bound estimate for . In this paper, we use the elementary method and some properties of the least solution of Pell’s equation to solve the conjecture partly. That is, we will prove the following.

Theorem 2. Equation (1) has no positive integer solution with and being odd primes.

2. Several Lemmas

Let be a nonsquare positive integer, and let denote the class number of binary quadratic primitive forms with discriminant . Then we have the following.

Lemma 3. For the equation there is a solution with , and there is a unique positive integer solution satisfying , where pass through all positive integer solutions of (2). We call as the least solution of (2). Every solution of (2) can be expressed as

Proof. See Section 10.9 of [6].

Lemma 4. Let be an odd prime satisfying and , and then every least solution of (2) satisfies .

Proof. See [7].

Lemma 5. Consider .

Proof. According to Table II in Chapter 16 of [6], we know that Lemma 5 holds for , and when , by Theorems and of [6], we have where is the least solution of (2). If , and , , then by (4), we have a contradiction. This proves Lemma 5.

Lemma 6. Let be an odd prime with . If the equation has the solution , then every solution can be expressed as where are positive integers satisfying and is a solution of (2).

Proof. See Lemma 4 of [8].

3. Proof of Theorem 2

Let be one of the solutions of (1). Without loss of generality we may assume that , since and are symmetrical in (1). By [5], we know that are coprime, and . When and are odd primes, must be even. Note that ; by (1) we have ; then , so by the result in [5], we get

By (9), we know that ; then from (1) we get . Therefore,

If , by (10), is an odd prime with . We see from (1) that the equation has the solution Since is an odd prime with , applying Lemma 6 to (11) and (12), we have where are positive integers satisfying is a solution of the equation and denotes the class number of binary quadratic primitive forms with discriminant .

Since is an odd prime, we know from (13) that or . If , by (13), so from (16) we have and . But, by Lemma 5, this is impossible; thus .

Since , by (13) we know that , so that (14) and (15) read

Further, , and from (19) we know , since by Lemma 3. Thus, according to Lemma 3 there is such that where is the least solution of (17). For the integer , there exist integers and satisfying Let From (17), (19), and (22), we know that and are integers satisfying And from (18), (20), (21), and (22), we have

If in (21), then, from (24), we have However, since from (9), and by (23), we have . According to (25), we get , which is impossible. Thus, from (21), we have and

Let Then are integers with , and where is the integral part of . From (29), we have Applying (27) to (24), we get From (17) and (26), , by (30), . However, we get from (9) that is an odd prime satisfying and ; then from Lemma 4, we know it is impossible. Thus, the theorem holds for .

Similarly, if , by (10) is an odd prime with . We see from (1) that the equation has the solution Applying Lemmas 5 and 6 to (11) and (12), we have where are positive integers satisfying and is a solution of the equation Applying Lemma 3 to (34) and (35), we have where is the least solution of (36). In addition, the integer can be expressed as Let From (35) and (36), we know that and are integers satisfying And from (34), (37), (38), and (39), we get

If in (38), then, from (41), we have Since is an odd prime, , by (42), we know From (40), we know ; then from (43) we get . Let , and , so By (45), Combining (42) and (46) we may immediately get However, from (42) and (47), we get , but it is impossible. Therefore, we have and

Now, using the similarly proof with , from (41) and (48) can obtain contradiction.

This completes the proof of our theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the P.S.F. (2013JZ001) and N.S.F. (11371291) of China. The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

References

  1. F. Luca and M. Mignotte, “On the equation yx±xy=z2,” The Rocky Mountain Journal of Mathematics, vol. 30, no. 2, pp. 651–661, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. Le, “On the Diophantine equation yx-xy=z2,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 4, pp. 1181–1185, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. Y. N. Liu and X. Y. Guo, “A diophantine equation and its integer solutions,” Acta Mathematica Sinica, vol. 53, no. 5, pp. 853–856, 2010. View at MathSciNet
  4. Z. Zhang and P. Yuan, “On the Diophantine equation axy+byz+czx=0,” International Journal of Number Theory, vol. 8, no. 3, pp. 813–821, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. Z.-F. Zhang, J.-G. Luo, and P.-Z. Yuan, “On the diophantine equation xy+yx=zz,” Chinese Annals of Mathematics, Series A, vol. 34A, no. 3, pp. 279–284, 2013.
  6. L.-G. Hua, An Introduction to Number Theory, Science Press, Beijing, China, 1979. View at MathSciNet
  7. A. J. van der Poorten, H. J. J. te Riele, and H. C. Williams, “Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than 100 000 000 000,” Mathematics of Computation, vol. 70, pp. 1311–1328, 2001. View at Publisher · View at Google Scholar
  8. M.-H. Le, “On the number of solutions to the Diophantine equation x2-D=pn,” Acta Mathematica Sinica, vol. 34, no. 3, pp. 378–387, 1991. View at MathSciNet