1 Introduction

Let a, b and c be coprime integers greater than one and let (x, y, z) be any positive integral solution of the equation \(a^x + b^y = c^z.\) In 2015, Hu and Le [2] proved that \(\max \{x, y, z\} \le \kappa (\log \max \{a, b, c\})^{3}\), where \(\kappa = 155{,}000\). Recently, the same authors improved the result as \(\kappa = 6500\), in [3]. In this paper, we prove the following theorem:

Theorem 1

For each \(0<\epsilon < 1\), there exists an effectively computable constant \(\kappa (\epsilon )> 0\) such that

$$\begin{aligned} \max \{x, y, z\} \le \kappa (\epsilon ) (\log \max \{a, b, c\})^{2 + \epsilon }. \end{aligned}$$

Moreover, if l, m and n are positive integers such that one of \((a\bmod 2, m, n)\), \((b\bmod 2, l, n)\) and \((c \bmod 2, l, m)\) is (0, 1, 1), then all positive integral solutions (x, y, z) of the equation

$$\begin{aligned} l a^x + m b^y = n c^z \end{aligned}$$

satisfy the inequality with \(\kappa (\epsilon ) = \kappa (\epsilon , l, m, n)\), an effectively computable constant depends only on \(\epsilon \), l, m and n.

2 Proof of Theorem 1

Though there are better lower bounds in the literature than what Lemma 1 gives, it is sufficient for the present purposes.

Lemma 1

(Corollary B.1 of Shorey-Tijdeman [4]) Let \(\alpha _1\), \(\alpha _2\), ..., \(\alpha _n\) be non-zero rational numbers of heights not exceeding \(A_1\), \(A_2\), ..., \(A_n\), respectively. We assume \(A_{j} \ge 3\) for \(1 \le j \le n\). Put

$$\begin{aligned} \Omega = \displaystyle \prod _{j = 1}^{n} \log A_{j} \,and \,\Omega ' = \displaystyle \prod _{j = 1}^{n-1} \log A_{j}. \end{aligned}$$

Then, there exist computable absolute constants \(\kappa _0\) and \(\kappa _1\) such that the inequalities

$$\begin{aligned} 0< \bigg |\alpha _{1}^{\beta _1}\alpha _{2}^{\beta _2} \cdots \alpha _{n}^{\beta _n} - 1 \bigg | < \exp \left( -(\kappa _0 n)^{\kappa _1 n}\Omega \log \Omega ' \log B \right) \end{aligned}$$

have no solution in rational integers \(\beta _1\), \(\beta _2\), ..., \(\beta _n\) of absolute values not exceeding \(B \ge 2\).

For a prime number p and a non-zero integer \(\alpha \), \({{\mathrm{ord}}}_{p}(\alpha )\) is the largest non-negative integer l such that \(p^l \mid \alpha \).

Lemma 2

(Bugeaud [1]) Let \(\alpha _1\) and \(\alpha _2\) be non-zero multiplicatively independent integers and let \(\beta _1\) and \(\beta _2\) be positive integers. Assume that there exist a positive integer g and a real number E such that

$$\begin{aligned} {{\mathrm{ord}}}_{p}\left( \alpha _{1}^{g} - 1\right) \ge E > 1/(p-1) \end{aligned}$$

and

$$\begin{aligned} {{\mathrm{ord}}}_{p}\left( \alpha _{2}^{g} - 1\right) > 0. \end{aligned}$$

Let \(A_1 >1\) and \(A_2 > 1\) be real numbers such that

$$\begin{aligned} \log A_i \ge \max \{\log |\alpha _i|, E \log p\} \end{aligned}$$

for \(i = 1, 2\). Put

$$\begin{aligned} \beta = \log \left( \frac{\beta _1}{\log A_2} + \frac{\beta _2}{\log A_1}\right) + \log (E \log p) + 0.4. \end{aligned}$$

Consider

$$\begin{aligned} \Lambda ' = \alpha _{1}^{\beta _1} - \alpha _{2}^{\beta _2}. \end{aligned}$$

Then, we have

$$\begin{aligned} {{\mathrm{ord}}}_p \Lambda ' \le \frac{36.1g}{E^3 (\log p)^{4}} \left( \max \{\beta , 6E \log p , 5\} \right) ^{2} (\log A_1) (\log A_2), \end{aligned}$$

if p is odd or if \(p =2\) and \({{\mathrm{ord}}}_{2}(\alpha _2 - 1) \ge 2\).

Fix \(\min \{x, y, z\} \ge 4\). First, we shall assume that c is even. Then a and b are odd. Here, we work modulo 4. Since \(a^x + b^y \equiv 0 \pmod {4}\) and so it is impossible that both x and y are even, \(\wedge ' = a^x + b^y\) can be written as \(\wedge ' = \alpha _{1}^{\beta _1} - \alpha _{2}^{\beta _2}\), where \(\alpha _{1}\), \(\alpha _{2}\), \(\beta _1\) and \(\beta _2\) are integers such that \(\alpha _1 \equiv \alpha _2 \equiv 1 \pmod {4}\) and \((\alpha _{1}, \alpha _{2}, \beta _1, \beta _2)\) is one of \((a, -b,x, y)\), \((b, -a, y,x)\), \((a^2,-b,x/2,y)\) and \((b^2,-a,y/2,x)\). Furthermore, take \(p=2\), \(E = 2\), \(g = 1\), \(A_1= \max \{4, |\alpha _{1}|\}\) and \(A_2 = \max \{4,|\alpha _2|\}\) in Lemma 2. Since \(\gcd (\alpha _1, \alpha _2) = 1\), by Lemma 2, we have

$$\begin{aligned} {{\mathrm{ord}}}_{2}\bigg (\alpha _{1}^{\beta _1} - \alpha _{2}^{\beta _2}\bigg ) < \kappa _{2}\left( \log |\alpha _{1}|\right) \left( \log |\alpha _{2}|\right) \left( \log \max \{\beta _1, \beta _2\}\right) ^2, \end{aligned}$$

where \(\kappa _{2}\) is a computable absolute constant. Also, it is clear that \(z \le {{\mathrm{ord}}}_{2}(c^{z})\). Therefore, we conclude that

$$\begin{aligned} z \le 4\kappa _2 (\log a)(\log b) \left( \log \max \{x, y\}\right) ^{2}. \end{aligned}$$

This provides the bound for \(\max \{x, y, z\}\), since \(a^x, b^y \le c^z\).

Next, assume that c is odd. Then a or b is even. Without loss of generality, suppose that a is even. Let \(\wedge '= c^z - b^y \). Then, similarly as above, we obtain

$$\begin{aligned} x \le 4\kappa _2 (\log b)(\log c) \left( \log \max \{y, z\}\right) ^2. \end{aligned}$$

Here, we distinguish two subcases. If \(c^z \le a^{2x}\), then the last two inequalities along with the inequality \(b^y \le c^z\) give the required bound. Otherwise, the given equation implies that

$$\begin{aligned} 0< c^z - b^y \le c^{z/2} \hbox { and so } 0 < 1 - c^{-z}b^y \le c^{-z/2}. \end{aligned}$$

Take \(n = 2\), \(\alpha _1 = c\), \(\alpha _2 = b\), \(\beta _1 = - z\), \(\beta _2 = y\), \(A_1 = \max \{3, c\}\), \(A_2 = \max \{3, b\}\) and \(B = \max \{y, z\}\) in Lemma 1. Therefore, by the lemma, we write

$$\begin{aligned} c^{-z/2} \ge 1 - c^{-z}b^y > \exp \{ -\kappa _{3}\left( \log b\right) \left( \log c\right) \left( \log \log \min \{b, c\}\right) \log \max \{y, z\}\}, \end{aligned}$$

where \(\kappa _{3}\) is a computable absolute constant. Combine this inequality with \(c^z \ge a^{2x}\) and \(b^y \le c^z\) to have the bound.

Next, fix \(\min \{x, y, z\} \le 3\). Then, \(a^x \le c^z\) and \(b^y \le c^z\) provide the required bound, if \(\min \{x, y, z\} = z\). Suppose that \(\min \{x, y, z\} = x\). Then, we get \(c^z - b^y \le a^3\) and it can be written as that \(0 < c^z b^{-y} - 1 \le a^3 b^{-y}\) if \(\max \{y, z\} = y\) or that \(0 < 1 - b^y c^{-z} \le a^3 c^{-z}\) if \(\max \{y, z\} = z\). Now, apply Lemma 1 by taking \(n = 2\), \(\alpha _1 = b\), \(\alpha _2 = c\), \(A_1 = \max \{3, b\}\), \(A_2 = \max \{3, c\}\) and \(B = \max \{y, z\}\). Then, we have the bound. Also, similarly as above, we can deal with the case \(\min \{x, y, z\} = y\).

For the proof of the second part of the theorem, in the above arguments replace \(c^z\) by \(nc^z\) if c is even, \(a^x\) by \(la^x\) if c is odd and a is even, and \(b^y\) by \(mb^y\) if c is odd and b is even, respectively.

Acknowlegements

I wish to remember my Ph.D. research supervisors Prof. R. Srikanth (SASTRA University) and Prof. R. Thangadurai (Harish-Chandra Research Institute) with gratitude. I express my sincere thanks to the referees of this paper for their valuable and constructive suggestions.