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Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$

According to this document, if we prove that $\pi$ has unbounded coefficients in its continued fraction expansion, then the answer to the above question is affirmative.

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  • $\begingroup$ What does $(a,b)=1$ mean? $\endgroup$
    – Nawaf Bou-Rabee
    Commented Jun 7, 2022 at 14:11
  • $\begingroup$ $\text{gcd}(a,b) = 1$ $\endgroup$
    – Siddharth Iyer
    Commented Jun 7, 2022 at 14:11

1 Answer 1

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I believe that this is an open problem. See e.g. the bottom of page 202 of this article by Bailey and Borwein. BTW the question was asked here before.

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