The Exponential Diophantine Equation 2^𝑥+𝑏^𝑦=𝑐^𝑧

Abstract

Let and be fixed coprime odd positive integers with . In this paper, a classification of all positive integer solutions of the equation is given. Further, by an elementary approach, we prove that if , then the equation has only the positive integer solution , except for and , where is a positive integer with .

1. Introduction

Let be the set of all positive integers. Let be fixed coprime positive integers with . In recent years, the solutions of the equation have been investigated in many papers (see [1–3] and its references). In this paper we deal with (1) for the case that . Then (1) can be rewritten as where and are fixed coprime odd positive integers with . We will give a classification of all solutions of (2) as follows.

Theorem 1. Every solution of (2) satisfies one of the following types:(i);(ii), where is a positive integer with ;(iii);(iv);(v) and ;(vi);(vii), , and ;(viii), , and ;(ix).
Recently, Miyazaki and Togbé [4] showed that if   and , then (2) has only the solution , except for . However, there are some exceptional cases missing from the result of [4]. In this paper, by an elementary approach, we prove the following result.

Corollary 2. If , then (2) has only the solution , except for and , where is a positive integer with .

2. Preliminaries

Lemma 3 (see [5, Formula 1.76]). For any positive integer and any complex numbers and , one has where is the integer part of ; are positive integers.

Lemma 4 (see [6]). Let and be coprime odd positive integers with . If the equation has solutions , then it has a unique solution such that , where through all solutions of (5). The solution is called the least solution of (5). Every solution of (5) can be expressed as Further, by (6), we have and .

Lemma 5. Equation (5) has no solutions such that , , and every prime divisor of and divides and , respectively.

Proof. We now assume that is a solution of (5) satisfying the hypothesis. Since , by Lemma 4, the is all solutions of (5). Let
We get where is odd. Numbers and are such that satisfy and satisfy . Thus, and are the odd indexed subsequences of the two Lehmer sequences of roots and . Their discriminants are and , respectively. Saying that all prime factors of divide implies that all primes of the th term of a Lehmer sequence divide its discriminant. The same is true for . Hence, and are terms of a Lehmer sequence of real roots lacking primitive divisors. By Table 2 in [7], this is possible only for . Even more, in the present case, is the th term of the Lucas sequence of positive real roots whose all prime factors divide its discriminant , and by Table 1 in [7] this is possible for odd only if or . Furthermore, when , we must have , but this is not possible since is not of the form for some positive integers , , and . So, only is possible. Now by some simple numerical computation for and , we see that it is not possible that all prime factors of and all prime factors of divide B. Thus, Lemma 5 is proved.

Lemma 6 (see [8]). The equation has only the solutions , and .

Lemma 7 (see [9]). Let be an odd positive integer with . If is a solution of the equation then .

Lemma 8 (see [10, 11]). The equation has only the solutions , and .

Lemma 9 (see [12]). The equation has only the solution .

Lemma 10 (see [13]). The equation has only the solution .

3. Proof of Theorem

Let be a solution of (2). If and , then we have , , and . It follows that Applying Lemma 10 to (15), we can only obtain the solutions of types (i) and (ii).

If and , then we have Applying Lemma 8 to (16), we can only get the solutions of types (iii), (iv), and (v).

Similarly, if and , using Lemmas 7 and 9, then we can only obtain the solutions of types (vi), (vii), and (viii). Finally, if , then the solutions are of type (ix). Thus, the theorem is proved.

4. Proof of Corollary

Since , (2) can be rewritten as Let be a solution of (17). By the theorem, (17) has only the solutions satisfying and .

If , and , then from (17) we get whence we obtain But, since and , congruence (20) is impossible.

If , and , by the theorem, then we have and . Hence, by (17), we get Since and , we see from (21) that . Substituting it into (17), we have and By (22), we get where is a solution of (10). Since and , by Lemma 6, we can only have and by (23).

If , then . Hence, by (17), we get and . It implies that the equation has the solution Notice that the least solution of (25) is ; and satisfy either or . Applying Lemma 5 to (26), we only obtain that and Thus, (17) has only the solutions (18), (24), and (27). The corollary is proved.

Conflict of Interests

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

Acknowledgments

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 N. S. F. (11371291) and P. N. S. F. (2013ZJ001) of China and G. I. C. F. (YZZ13075) of NWU.