Let $p$ be a positive real number. For any fixed $\epsilon>0$ does there exist a positive integer $n$ such that fractional part of $p^n$ is less than $\epsilon$?
Add-on: $p$ is rational. (original qn I had in mind).
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$\begingroup$ This doesn't seem like a bad question to me. Perhaps those voting to close could say what the answer is? $\endgroup$– LuciaCommented Dec 6, 2013 at 5:11
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4$\begingroup$ The answer is no. Take $p = 2 + \sqrt{2}$ and use the fact that $(2 + \sqrt{2})^n + (2 - \sqrt{2})^n$ is an integer. More generally see en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number . $\endgroup$– Qiaochu YuanCommented Dec 6, 2013 at 5:18
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$\begingroup$ @QiaochuYuan: Yes indeed; I was thinking distance from the nearest integer (which admittedly was not what the OP asked). Any idea then? $\endgroup$– LuciaCommented Dec 6, 2013 at 5:20
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$\begingroup$ forgive me for my bad phrasing. Thanks for the answer $\endgroup$– 61plusCommented Dec 6, 2013 at 5:28
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1$\begingroup$ For $p$ rational some googling (infres.enst.fr/~jsaka/PUB/Files/RBNS-rev.pdf) suggests this is an open problem. $\endgroup$– Qiaochu YuanCommented Dec 6, 2013 at 7:43
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