Pillai's conjecture -- that the gap between (nontrivial) powers is unbounded below -- is still open (it would be a consequence of the $abc$ conjecture, were that proven). But I wonder what the right order of magnitude for it is, even though a proof seems far off.


Suppose $n=|a^x-b^y|$ for integers $a,b,x,y$ with $a,b\ge1$ and $x,y\ge2.$ What is the (conjectural) maximal order of $a^x$?


$17 = 378661^2 - 5234^3$ so we should not expect it to be too tiny. But I don't even know if the right order should be polynomial, exponential, doubly-exponential, etc. Does anyone have insight here?



Maximum size of powers with a given difference
[Perfect Powers: Pillai's works and their developments](http://arxiv.org/abs/0908.4031)
is related to your question.


p.9:
Conjecture 3.1. For any $\epsilon > 0$, there exists a constant $\kappa(\epsilon) > 0$ such that, for any positive integers $(a, b, x, y)$, with $x \ge 2, y \ge 2$ and $a^x \ne b^y$ ,
$$|a^x - b^y | \ge \kappa(\epsilon) \max (a^x , b^y)^{ 1-(1/x)-(1/y)-\epsilon}$$


p.7
In 1986, J. Turk [126] gave an effective estimate from below for $|x^n - y^m |$, which was improved by B. Brindza, J.-H. Evertse and K. Gyory in 1991 and further refined by Y. Bugeaud in 1996:


Let $x$ be a positive integer and $y, n, m$ be integers which are $\ge 2$. Assume $x^n \ne y^m$ . Then $|x^n - y^m | \ge m^{2/(5n)} n^{-5} 2^{-6-42/n}.$ 




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*Added later* There are unconditional [results for the abc conjecture](https://en.wikipedia.org/wiki/Abc_conjecture#Theoretical_results) which
give you some quite large bounds.


