The Scientific World Journal
Volume 2015 (2015), Article ID 306590, 3 pages
http://dx.doi.org/10.1155/2015/306590

Research Article

The Diophantine Equation

Lan Qi¹ and Xiaoxue Li²

¹College of Mathematics and Statistics, Yulin University, Yulin, Shaanxi 719000, China
²School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Received 7 July 2014; Accepted 14 December 2014

Academic Editor: Abdelghani Bellouquid

Copyright © 2015 Lan Qi 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.

Abstract.

Let be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if , then the equation has no positive integer solutions ; (ii) if , then the equation has only the solutions , where is an odd prime with ; (iii) if and , then the equation has at most two positive integer solutions .

1. Introduction

Let , be the sets of all integers and positive integers, respectively. Let be a fixed odd prime. Recently, the solutions of the equation

were determined in the following cases:

(1)(Sroysang [1]) if , then (1) has no solutions;(2)(Sroysang [2]) if , then (1) has no solutions;(3)(Rabago [3]) if , then (1) has only the solutions , , and .

In this paper, using certain results of exponential Diophantine equations, we prove a general result as follows.

Theorem 1. If , then (1) has no solutions . If , then (1) has only the solutions

where is an odd prime with .
If and , then (1) has at most two solutions .

Obviously, the above theorem contains the results of [1, 2]. Finally, we propose the following conjecture.

Conjecture 2. If , then (1) has at most one solution .

2. Preliminaries

Lemma 3. If is a prime, where is a positive integer, then must be a prime.

Proof. See Theorem of [4].

Lemma 4. If is an odd prime with , then the equation

has solutions .

Proof. See Section 8.1 of [5].

Lemma 5. The equation

has only the solution .

Proof. See Theorem 8.4 of [6].

Lemma 6. Let be a fixed odd positive integer. If the equation

has solutions , then the equation

has at most two solutions , except the following cases:

(i), , , , and , where is a positive integer with ;(ii), , , and , where is a positive integer with ;(iii), , , and , where , are positive integers with .

Proof. See [7].

Lemma 7. If is an odd prime and belongs to the exceptional case (i) of Lemma 6, then .

Proof. We now assume that is an odd prime with . Then we have

If , since , then , and by (7), we have

But, by the second equality of (9), we get , a contradiction.
If , then from (8) we get

Further, by the second equality of (10), we have , , and . Thus, the lemma is proved.

Lemma 8. If is an odd prime and belongs to the exceptional case (iii) of Lemma 6, then .

Proof. Using the same method as in the proof of Lemma 7, we can obtain this lemma without any difficulty.

Lemma 9. If belongs to the exceptional case (ii), then (6) has at most one solution with .

Proof. Notice that, for any positive integer , there exists at most one number of 5, , and which is a multiple of 3. Thus, by Lemma 6, the lemma is proved.

Lemma 10. The equation

has only the solution .

Proof. See [8].

3. Proof of Theorem

We now assume that is a solution of (1). Then we have .

If , since , then from (1) we get

where we obtain

Since , applying Lemma 10 to (14), we get

Further, by Lemma 3, we see from the second equality of (15) that

is an odd prime with .

Therefore, by (13), (15), and (16), we obtain the solutions given in (2).

Obviously, if satisfies (2), then . Otherwise, since , we see from (1) that . It implies that if , then (1) has no solutions . If , then (1) has only the solutions (2).

Here and below, we consider the remaining cases that . By the above analysis, we have . If , then and (4) has the solution with . But, by Lemma 5, it is impossible. Therefore, we have

Substituting (17) into (1), the equation

has the solution with . Since , by Lemma 4, (3) has solutions . Therefore, by Lemmas 6–9, (1) has at most two solutions . Thus, the theorem is proved.

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11371291).

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