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Here I ask a question concerning the diophantine equations $$x^n+n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\},\tag{1}$$ and $$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$

QUESTION: Besides the two solutions $$5^2+2=3^3\quad\text{and}\quad{5^3+3=2^7},\tag{3}$$ does the equation $(1)$ have other solutions? Besides the two solutions $$2^5-5=3^3\quad\text{and}\quad{2^7-7=11^2},\tag{4}$$ does the equation $(2)$ have other solutions?

Actually I formulated this question in 2013, and conjectured that the equation $(1)$ only has two solutions as stated in $(3)$ and the equation $(2)$ only has two solutions as stated in $(4)$. See my preprint http://arxiv.org/abs/1312.1166.

The equations $(1)$ and $(2)$ are similar to Catalan's equation $x^n-y^m=1$ with $m,n,x,y\in\{2,3,\ldots\}$. Any helpful ideas concerning my question?

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    $\begingroup$ Why do you think this diophantine equation is interesting? $\endgroup$
    – Dmitry Vaintrob
    Commented Jul 17, 2018 at 5:12
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    $\begingroup$ Finiteness of solutions follows of course from the $abc$ conjecture. They look completely unassailable to me; Catalan is very special, and just changing the ``$1$'' to $2$ or $n$ or $m$ leads to equations apparently beyond all known techniques. $\endgroup$
    – Vesselin Dimitrov
    Commented Jul 17, 2018 at 7:50
  • $\begingroup$ In the cited arXiv paper, I showed that for any integer $x > 1$, the sets $\{x^n+n:\ n=1,2,3,\ldots\}$ and $\{x^n-n:\ n=1,2,3,\ldots\}$ contain a complete system of residues modulo any positive integer. So I consider numbers of the form $x^n\pm n$ interesting and ask a question similar to Catalan's conjecture. $\endgroup$
    – Zhi-Wei Sun
    Commented Jul 18, 2018 at 2:42

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