In this post I present my variations of the problem involving Nagell-Ljunggren equation, that is explained in pages 10 and 11 of *Highlights in the Research Work of T. N. Shorey* by R. Tijdeman, from *Diophantine Equations*, (Editor) N. Saradha, Tata Institute of Fundamental Research, Narosa Publishing House (2008).


Consider for integers $x>1$, $1<y<z$ and integers $1<p$ and $2<n$ the equations $$y^p+z^p=(y^{p-1}+z^{p-1})\cdot\frac{x^n-1}{x-1},\tag{1}$$
and $$2yz=(y+z)\cdot\frac{x^n-1}{x-1}.\tag{2}$$


**Conjecture 1.** *Consider the equation* $(2)$ *over integers* $x>1$, $1<y<z$ *and* $2<n$, *then there exists a positive integer* $n\_0\geq 8$ *such that* $\forall n>n\_0$ *this equation has no solutions.*


**Conjecture 2.** *We consider the equation* $(1)$ *over integers* $x>1$, $1<y<z$, $1<p$, *and* $2<n$ *then there exists a positive constant* $C$ *such that the equation has no solutions whenever* $n\cdot p>C$.


Both equations ask for Lehmer means $L\_p(y,z)$ with all the digits equal to $1$ in the corresponding $x-$adic expansion


$$L\_p(y,z)=1+x+\ldots+x^{n-1}.$$


Wikipedia has an article for [Lehmer mean,](https://en.wikipedia.org/wiki/Lehmer_mean) and their relationship to the harmonic mean.



> 
> **Question.** I would like to know if it is possible (what work can be done about it) to **prove or disprove** Conjecture 1 and/or Conjecture 2. **Many thanks.**
> 
> 
> 


I don't know if preveious problems are in the literature, I hope that these problems have a good mathematical content and are interesting for you. I'm asking about what work can be done about the Conjecture 1 or well about the Conjecture 2. Finally I add two solutions from the following examples.


**Example 1.** One has that $(x,y,z;n,p)=(16,225,300;3,3)$ solves $(1)$ because $$\frac{225^3+300^3}{225^2+300^2}=272=1+1\cdot 16+1\cdot 16^2.$$


**Example 2.** Similarly $(x,y,z;n)=(11,91,247;3)$ solves $(2)$ because $$\frac{2}{\frac{1}{91}+\frac{1}{247}}=133=1+1\cdot 11+1\cdot 11^2.$$



Diophantine equations that involve Lehmer means with all digits equal to $1$ in their $x-$adic expansions
Conjecture 1 does not hold as for any even $n$, (2) has a solution:
$$(x,y,z)=(2,\frac23(2^n-1),2(2^n-1))$$


Likewise, Conjecture 2 fails as for $p=2$ and any even $n$, (1) has a solution:
$$(x,y,z)=(4,\frac2{15}(4^n-1),\frac25(4^n-1))$$


