Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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...

share|improve this question
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
Note that the question asks for all $c>0$, not for some $c>0$. –  Emil Jeřábek Jun 11 '12 at 15:05
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.