TITLE: $a-b,a^2-b^2,a^3-b^3...$ are integers $\implies$ $a,b$ are integers?
QUESTION [8 upvotes]: Let $a,b$ be distinct real numbers such that $a^n -b^n$ is integer for every positive integer $n$ , then is it true that $a,b$ are integers ?

REPLY [2 votes]: This answer is really just the same as barto's, but written up in a more self contained fashion. We may write, as barto has already shown,  $a = \frac{c}{d}, b = \frac{e}{d}$ with $ce$ relatively prime to $d$ and $c,d,e$ integers with $d >0.$ Now for every $d,$ we have $c^{n} \equiv e^{n}$ (mod $d^{n}$). Suppose that $d > 1.$ Then for every integer $n>0,$ we have $c^{n}-e^{n} = (c-e) \sum_{j=0}^{n-1}c^{j}e^{n-1-j}.$ Hence $\frac{c^{n}-e^{n}}{c-e} \equiv nc^{n-1}$ (mod $d$) But $c$ is relatively prime to $d,$ so whenever $n$ is relatively prime to $d,$ we conclude that $\frac{c^{n}-e^{n}}{c-e}$ is relatively prime to $d$. Hence $c-e$ is divisible by $d^{n},$ as $c^{n}-e^{n}$ is, a contradiction, as $d >1$  and $n$ can be arbitrarily large.