# A note on the equation ${x}^{y}+{y}^{z}={z}^{x}$

## Abstract

In this paper, we shall use some simple inequalities and a deep result on the existence of primitive divisors of Lucas numbers to prove that the exponential Diophantine equation ${x}^{y}+{y}^{z}={z}^{x}$ has no positive integer solution $\left(x,y,z\right)$ with $2\mid y$.

MSC:11D61.

## 1 Introduction

Let , be the sets of all integers and positive integers, respectively. Recently, Zhang and Yuan [1] were interested in the equation

${x}^{y}+{y}^{z}={z}^{x},\phantom{\rule{1em}{0ex}}x,y,z\in \mathbb{N}.$
(1.1)

Using the Gel’fond-Baker method, they proved that all solutions $\left(x,y,z\right)$ of (1.1) satisfy $max\left\{x,y,z\right\}. This upper bound is far beyond the computable scope at present. In this paper, we shall use some simple inequalities and a deep result on the existence of primitive divisors of Lucas numbers to prove the following result.

Theorem Equation (1.1) has no solution $\left(x,y,z\right)$ with $2\mid y$.

In addition, it is obvious that $\left(x,y,z\right)=\left(1,1,2\right)$ is a solution of (1.1). Because one have not found the other solutions, we propose a conjecture as follows:

Conjecture Equation (1.1) has only the solution $\left(x,y,z\right)=\left(1,1,2\right)$.

Our theorem supports the above mentioned conjecture.

## 2 Preliminaries

Lemma 2.1 Let $f\left(X\right)=X/logX$, where X is a real number. Then $f\left(X\right)$ is an increasing function for $X>e$.

Proof Since ${f}^{\prime }\left(X\right)=\left(logX-1\right)/{\left(logX\right)}^{2}$, we have ${f}^{\prime }\left(X\right)>0$ for $X>e$. Thus, the lemma is proved. □

Lemma 2.2 Let $g\left(X\right)=\sqrt{X}-2\left(2+log\left(4X\right)\right)/\pi$, where X is a real number. Then we have $g\left(X\right)>0$ for $X\ge 16$.

Proof Since ${g}^{\prime }\left(X\right)=1/2\sqrt{X}-2/\pi X>0$ for $X\ge 16$, g(X) is an increasing function satisfying $g\left(X\right)\ge g\left(16\right)>0$ for $X\ge 16$. The lemma is proved. □

Lemma 2.3 ([2, 3])

The equation

${X}^{2}+{2}^{m}={Y}^{n},\phantom{\rule{1em}{0ex}}X,Y,m,n\in \mathbb{N},gcd\left(X,Y\right)=1,n>2$
(2.1)

has only the solutions $\left(X,Y,m,n\right)=\left(5,3,1,3\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}\left(7,3,5,4\right)$.

Lemma 2.4 ([[4], Theorem 8.4])

The equation

${X}^{2}+{Y}^{m}={2}^{n},\phantom{\rule{1em}{0ex}}X,Y,m,n\in \mathbb{N},2\nmid Y,Y>1,m>1$
(2.2)

has only the solution $\left(X,Y,m,n\right)=\left(13,7,3,9\right)$.

Lemma 2.5 ([[4], Theorem 8.4])

The equation

${X}^{2}-{Y}^{m}={2}^{n},\phantom{\rule{1em}{0ex}}X,Y,m,n\in \mathbb{N},2\nmid Y,Y>1,m>2,n>1$
(2.3)

has only the solution $\left(X,Y,m,n\right)=\left(71,17,3,7\right)$.

Let D be a positive integer, and let $h\left(-4D\right)$ denote the class number of positive binary quadratic primitive forms of discriminant $-4D$.

Lemma 2.6 $h\left(-4D\right)\le D$.

Proof Notice that $h\left(-4\right)=h\left(-8\right)=h\left(-28\right)=1$, $h\left(-12\right)=h\left(-16\right)=h\left(-20\right)=h\left(-24\right)=h\left(-32\right)=h\left(-36\right)=h\left(-40\right)=h\left(-52\right)=h\left(-60\right)=2$, $h\left(-44\right)=3$, $h\left(-48\right)=h\left(-56\right)=4$. The lemma holds for $D\le 15$. By Theorems 11.4.3, 12.10.1, and 12.14.3 of [5], if $D\ge 1$, then

$h\left(-4D\right)<\frac{2\sqrt{D}}{\pi }\left(2+log\left(4D\right)\right).$
(2.4)

Therefore, if $h\left(-4D\right)>D$, then from (2.4) we get

$\sqrt{D}<\frac{2}{\pi }\left(2+log\left(4D\right)\right).$
(2.5)

But, by Lemma 2.2, (2.5) is impossible for $D\ge 16$. Thus, the lemma is proved. □

Lemma 2.7 Let k be a positive integer with $gcd\left(k,2D\right)=1$. Every solution $\left(X,Y,Z\right)$ of the equation

${X}^{2}+D{Y}^{2}={k}^{Z},\phantom{\rule{1em}{0ex}}X,Y,Z\in \mathbb{Z},gcd\left(X,Y\right)=1,Z>0,$
(2.6)

can be expressed as

$\begin{array}{c}Z={Z}_{1}t,\phantom{\rule{1em}{0ex}}t\in \mathbb{N},\hfill \\ X+Y\sqrt{-D}={\lambda }_{1}{\left({X}_{1}+{\lambda }_{2}{Y}_{1}\sqrt{-D}\right)}^{t},\phantom{\rule{1em}{0ex}}{\lambda }_{1},{\lambda }_{2}\in \left\{±1\right\},\hfill \end{array}$

where ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$ are positive integers satisfying

${X}_{1}^{2}+D{Y}_{1}^{2}={k}^{{Z}_{1}},\phantom{\rule{1em}{0ex}}gcd\left({X}_{1},{Y}_{1}\right)=1,{Z}_{1}\mid h\left(-4D\right).$

Proof This lemma is the special case of [[6], Theorems 1 and 2] for ${D}_{1}=1$ and ${D}_{2}<0$.

Let α, β be algebraic integers. If $\alpha +\beta$ and αβ are nonzero coprime integers and $\alpha /\beta$ is not a root of unity, then $\left(\alpha ,\beta \right)$ is called a Lucas pair. Further, let $a=\alpha +\beta$ and $c=\alpha \beta$. Then we have

$\alpha =\frac{1}{2}\left(a+\lambda \sqrt{b}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\beta =\frac{1}{2}\left(a-\lambda \sqrt{b}\right),\phantom{\rule{1em}{0ex}}\lambda \in \left\{±1\right\},$

where $b={a}^{2}-4c$. We call $\left(a,b\right)$ the parameters of the Lucas pair $\left(\alpha ,\beta \right)$. Two Lucas pairs $\left({\alpha }_{1},{\beta }_{1}\right)$ and $\left({\alpha }_{2},{\beta }_{2}\right)$ are equivalent if ${\alpha }_{1}/{\alpha }_{2}={\beta }_{1}/{\beta }_{2}=±1$. Given a Lucas pair $\left(\alpha ,\beta \right)$, one defines the corresponding sequence of Lucas numbers by

${L}_{n}\left(\alpha ,\beta \right)=\frac{{\alpha }^{n}-{\beta }^{n}}{\alpha -\beta },\phantom{\rule{1em}{0ex}}n=0,1,2,\dots .$

For equivalent Lucas pairs $\left({\alpha }_{1},{\beta }_{1}\right)$ and $\left({\alpha }_{2},{\beta }_{2}\right)$, we have ${L}_{n}\left({\alpha }_{1},{\beta }_{1}\right)=±{L}_{n}\left({\alpha }_{2},{\beta }_{2}\right)$ for any $n\ge 0$. A prime p is called a primitive divisor of ${L}_{n}\left(\alpha ,\beta \right)$ ($n>1$) if $p\mid {L}_{n}\left(\alpha ,\beta \right)$ and $p\nmid b{L}_{1}\left(\alpha ,\beta \right)\cdots {L}_{n-1}\left(\alpha ,\beta \right)$. A Lucas pair $\left(\alpha ,\beta \right)$ such that ${L}_{n}\left(\alpha ,\beta \right)$ has no primitive divisor will be called an n-defective Lucas pair. Further, a positive integer n is called totally non-defective if no Lucas pair is n-defective. □

Lemma 2.8 ([7])

Let n satisfy $4 and $n\ne 6$. Then, up to equivalence, all parameters of n-defective Lucas pairs are given as follows:

1. (i)

$n=5$, $\left(a,b\right)=\left(1,5\right),\left(1,-7\right),\left(2,-40\right),\left(1,-11\right),\left(1,-15\right),\left(12,-76\right),\left(12,-1,364\right)$.

2. (ii)

$n=7$, $\left(a,b\right)=\left(1,-7\right),\left(1,-19\right)$.

3. (iii)

$n=8$, $\left(a,b\right)=\left(2,-24\right),\left(1,-7\right)$.

4. (iv)

$n=10$, $\left(a,b\right)=\left(2,-8\right),\left(5,-3\right),\left(5,-47\right)$.

5. (v)

$n=12$, $\left(a,b\right)=\left(1,5\right),\left(1,-7\right),\left(1,-11\right),\left(2,-56\right),\left(1,-15\right),\left(1,-19\right)$.

6. (vi)

$n\in \left\{13,18,30\right\}$, $\left(a,b\right)=\left(1,-7\right)$.

Lemma 2.9 ([8])

If $n>30$, then n is totally non-defective.

## 3 Further lemmas on the solutions of (1.1)

Throughout this section, we assume that $\left(x,y,z\right)$ is a solution of (1.1) with $\left(x,y,z\right)\ne \left(1,1,2\right)$.

Lemma 3.1 ([1])

x, y and z are coprime.

Lemma 3.2 $min\left\{x,y,z\right\}\ge 3$.

Proof Since ${z}^{x}={x}^{y}+{y}^{z}>1$, we have $z>1$. If $x=1$, since $\left(x,y,z\right)\ne \left(1,1,2\right)$, then $y>1$ and $z=1+{y}^{z}\ge 1+{2}^{z}\ge z+3$, a contradiction. Similarly, if $y=1$, then $x>1$ and $x+1={z}^{x}\ge {2}^{x}\ge x+2$, a contradiction. Therefore, we have $min\left\{x,y,z\right\}\ge 2$.

If $x=2$, then

${2}^{y}+{y}^{z}={z}^{2}.$
(3.1)

Further, by Lemma 3.1, y and z are odd integers with $min\left\{y,z\right\}\ge 3$. Hence, we see from (3.1) that (2.3) has the solution $\left(X,Y,m,n\right)=\left(z,y,z,y\right)$. But, by Lemma 2.5, it is impossible.

Similarly, if $y=2$ or $z=2$, then we have

${x}^{2}+{2}^{z}={z}^{x},\phantom{\rule{1em}{0ex}}2\nmid xz,min\left\{x,z\right\}\ge 3$
(3.2)

or

${x}^{y}+{y}^{2}={2}^{x},\phantom{\rule{1em}{0ex}}2\nmid xy,min\left\{x,y\right\}\ge 3.$
(3.3)

But, by Lemmas 2.3 and 2.4, (3.2) and (3.3) are impossible. Thus, we get $min\left\{x,y,z\right\}\ge 3$. The lemma is proved. □

Lemma 3.3 $y.

Proof By (1.1), we have ${z}^{x}>{x}^{y}$ and ${z}^{x}>{y}^{z}$. Hence,

$\frac{x}{logx}>\frac{y}{logz}$
(3.4)

and

$\frac{x}{logy}>\frac{z}{logz}.$
(3.5)

In addition, by Lemmas 3.1 and 3.2, x, y and z are distinct.

If $x, by Lemma 3.2, then $3\le x. Hence, by Lemma 2.1, we get

$\frac{z}{logz}>\frac{x}{logx}>\frac{x}{logy},$
(3.6)

which contradicts (3.5). Similarly, we can remove the case that $x.

If $z, then $3\le z and

$\frac{y}{logz}>\frac{y}{logy}>\frac{x}{logx},$
(3.7)

which contradicts (3.4). Thus, we get $y. The lemma is proved. □

## 4 Proof of theorem

We now assume that $\left(x,y,z\right)$ is a solution of (1.1) with $2\mid y$. Since $\left(x,y,z\right)\ne \left(1,1,2\right)$, by Lemmas 3.1, 3.2 and 3.3, we have $2\nmid xz$, $gcd\left(y,z\right)=1$, $min\left\{x,y,z\right\}\ge 3$ and $x>y$.

We see from (1.1) that the equation

${X}^{2}+y{Y}^{2}={z}^{Z},\phantom{\rule{1em}{0ex}}X,Y,Z\in \mathbb{Z},gcd\left(X,Y\right)=1,Z>0$
(4.1)

has the solution

$\left(X,Y,Z\right)=\left({x}^{y/2},{y}^{\left(z-1\right)/2},x\right).$
(4.2)

Applying Lemma 2.7 to (4.1) and (4.2), we have

$x={Z}_{1}t,\phantom{\rule{1em}{0ex}}t\in \mathbb{N},$
(4.3)
${x}^{y/2}+{y}^{\left(z-1\right)/2}\sqrt{-y}={\lambda }_{1}{\left({X}_{1}+{\lambda }_{2}{Y}_{1}\sqrt{-y}\right)}^{t},\phantom{\rule{1em}{0ex}}{\lambda }_{1},{\lambda }_{2}\in \left\{±1\right\},$
(4.4)

where ${X}_{1}$, ${Y}_{1}$, ${Z}_{1}$ are positive integers satisfying

${X}_{1}^{2}+y{Y}_{1}^{2}={z}^{{Z}_{1}},\phantom{\rule{1em}{0ex}}gcd\left({X}_{1},{Y}_{1}\right)=1$
(4.5)

and

${Z}_{1}\mid h\left(-4y\right).$
(4.6)

Let

$\alpha ={X}_{1}+{Y}_{1}\sqrt{-y},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\beta ={X}_{1}-{Y}_{1}\sqrt{-y}.$
(4.7)

We see from (4.5) and (4.7) that $\alpha +\beta =2{X}_{1}$ and $\alpha \beta ={z}^{{Z}_{1}}$ are coprime nonzero integers, $\alpha /\beta =\left(\left({X}_{1}^{2}-y{Y}_{1}^{2}\right)+2{X}_{1}{Y}_{1}\sqrt{-y}\right)/{z}^{{Z}_{1}}$ is not a root of unity. Hence, $\left(\alpha ,\beta \right)$ is a Lucas pair with parameters $\left(2{X}_{1},-4y{Y}_{1}^{2}\right)$. Further, Let ${L}_{n}\left(\alpha ,\beta \right)$ ($n=0,1,2,\dots$) denote the corresponding Lucas numbers. By (4.4) and (4.7), we have

${y}^{\left(z-1\right)/2}=|{L}_{t}\left(\alpha ,\beta \right)|.$
(4.8)

We find from (4.7) and (4.8) that the Lucas number ${L}_{t}\left(\alpha ,\beta \right)$ has no primitive divisor. Therefore, by Lemma 2.9, we have $t\le 30$. Further, since $2\nmid x$ and $2\nmid t$ by (4.3), it is easy to remove all cases in Lemma 2.8 and conclude that $t\in \left\{1,3\right\}$.

If $t=3$, then from (4.4) we get

${y}^{\left(z-1\right)/2}={\lambda }_{1}{\lambda }_{2}{Y}_{1}\left(3{X}_{1}^{2}-y{Y}_{1}^{2}\right).$
(4.9)

Let $d=gcd\left({Y}_{1},3{X}_{1}^{2}-y{Y}_{1}^{2}\right)$. Since $gcd\left({X}_{1},{Y}_{1}\right)=1$, we have $d\mid 3$ and $d\in \left\{1,3\right\}$. Further, since $t\mid x$, we get $3\mid x$, $3\nmid y$ and $d\ne 3$ by (4.9). Therefore, we have $d=1$ and, by (4.9), $gcd\left(y,3{X}_{1}^{2}-y{Y}_{1}^{2}\right)=1$ and

${Y}_{1}={y}^{\left(z-1\right)/2},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}3{X}_{1}^{2}-y{Y}_{1}^{2}=±1.$
(4.10)

It implies that

$3{X}_{1}^{2}\mp 1={y}^{z}.$
(4.11)

But, since $2\mid y$ and $z\ge 3$, we get from (4.11) that $2\nmid {X}_{1}$ and $0\equiv {y}^{z}\equiv 3{X}_{1}^{2}\mp 1\equiv 3\mp 1\not\equiv 0\left(mod8\right)$, a contradiction.

If $t=1$, then from (4.3) and (4.6) that $x={Z}_{1}$, $x\mid h\left(-4y\right)$ and

$x\le h\left(-4y\right).$
(4.12)

But recall that $x>y$, by Lemma 2.6, (4.12) is impossible. Thus, (1.1) has no solution $\left(x,y,z\right)$ with $2\mid y$. The theorem is proved.

## References

1. Zhang Z-F, Yuan P-Z:On the diophantine equation $a{x}^{y}+b{y}^{z}+c{z}^{x}=0$. Int. J. Number Theory 2012,8(3):813-821. 10.1142/S1793042112500467

2. Cohn JHE:The diophantine equation ${x}^{2}+{2}^{k}={y}^{n}$. Arch. Math. Basel 1992,59(3):341-343.

3. Le M-H:On Cohn’s conjecture concerning the diophantine equation ${x}^{2}+{2}^{m}={y}^{n}$. Arch. Math. Basel 2002,78(1):26-35. 10.1007/s00013-002-8213-5

4. Bennett MA, Skinner CM: Ternary diophantine equations via Galois representations and modular forms. Can. J. Math. 2004,56(1):23-54. 10.4153/CJM-2004-002-2

5. Hua L-K: Introduction to Number Theory. Springer, Berlin; 1982.

6. Le M-H:Some exponential diophantine equations I: the equation ${D}_{1}{x}^{2}-{D}_{2}{y}^{2}=\lambda {k}^{z}$. J. Number Theory 1995,55(2):209-221. 10.1006/jnth.1995.1138

7. Voutier PM: Primitive divisors of Lucas and Lehmer sequences. Math. Comput. 1995, 64: 869-888. 10.1090/S0025-5718-1995-1284673-6

8. Bilu Y, Hanrot G, Voutier PM: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 2001, 539: 75-122. (with an appendix by M Mignotte)

## Acknowledgements

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 the P. S. F. (2013JZ001) and N.S.F. (11371291) of P.R. China.

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Correspondence to Xiaoxue Li.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

YL obtained the theorems and completed the proof. XL corrected and improved the final version. Both authors read and approved the final manuscript.

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Lu, Y., Li, X. A note on the equation ${x}^{y}+{y}^{z}={z}^{x}$. J Inequal Appl 2014, 170 (2014). https://doi.org/10.1186/1029-242X-2014-170

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• DOI: https://doi.org/10.1186/1029-242X-2014-170