#### 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).