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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Small fractional part of powers of a real number [closed]

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 q …
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