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Is it known whether, for all $c > 0$, there always exist integers $p$ and $q$ such that $\left| \pi - \frac{p}{q}\right| < \frac{c}{q^2}$?

This seems like a fundamental question but I couldn't find a reference...

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1  
yes: mathoverflow.net/questions/53724/… – Alex R. Jun 11 '12 at 14:38
    
This is a basic result from continued fractions. Any intro number theory text that has a section on cf's will contain this result. – Kevin O'Bryant Jun 11 '12 at 14:49
    
@Kevin: It may be just my ignorance in the subject, but it would seem to me that the statement the OP wants would require the terms of the cf of $\pi$ to be unbounded, and this seems to be an open problem. – Emil Jeřábek Jun 11 '12 at 14:59
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Note that the question asks for all $c>0$, not for some $c>0$. – Emil Jeřábek Jun 11 '12 at 15:05
3  
mathworld.wolfram.com/PiContinuedFraction.html gives some information about this question, which seems to confirm that this is open. – Lee Mosher Jun 11 '12 at 15:40

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