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

Yuanyuan Deng and Wenpeng Zhang

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 [1–4]). 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.

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