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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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    $\begingroup$ yes: mathoverflow.net/questions/53724/… $\endgroup$
    – Alex R.
    Commented Jun 11, 2012 at 14:38
  • $\begingroup$ This is a basic result from continued fractions. Any intro number theory text that has a section on cf's will contain this result. $\endgroup$
    – Kevin O'Bryant
    Commented Jun 11, 2012 at 14:49
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    $\begingroup$ Note that the question asks for all $c>0$, not for some $c>0$. $\endgroup$
    – Emil Jeřábek
    Commented Jun 11, 2012 at 15:05
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    $\begingroup$ @Emil: you're right; I missed the "for all c" clause. $\endgroup$
    – Kevin O'Bryant
    Commented Jun 11, 2012 at 15:24
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    $\begingroup$ mathworld.wolfram.com/PiContinuedFraction.html gives some information about this question, which seems to confirm that this is open. $\endgroup$
    – Lee Mosher
    Commented Jun 11, 2012 at 15:40

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