query: Find two rational numbers $\frac ab$ satisfying $\mid \pi - \frac ab\mid < \frac{1}{\sqrt 5 b^2}.$
positive_passages.docid: 153831
positive_passages.text: Approximating $\pi$ with least digits
negative_passages.docid: 20834
negative_passages.text: Integer solutions to $c=a^2-b^2$
negative_passages.docid: 77470
negative_passages.text: Find modulus of $\frac{|z_1-z_2|}{|1-(z_1)(\overline{z_2})|}$
negative_passages.docid: 39492
negative_passages.text: Finding $z\in \mathbb C$ with $\sin z = 2$?
negative_passages.docid: 24499
negative_passages.text: Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$
negative_passages.docid: 56802
negative_passages.text: Why $\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}$ is a rational number?
negative_passages.docid: 100336
negative_passages.text: Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
negative_passages.docid: 132931
negative_passages.text: Determining the parity of the coefficients of a quadratic given that there is a rational solution
negative_passages.docid: 68389
negative_passages.text: Proving $ \frac{(a-b)^2}{8a}\leqslant\frac{a+b}{2} - \sqrt{ab} \leqslant\frac{(a-b)^2}{8b}$ when $ 0<b\le a$
negative_passages.docid: 147410
negative_passages.text: ArcTan(2) a rational multiple of $\pi$?
negative_passages.docid: 150037
negative_passages.text: Finding the simplest rational in a closed interval
negative_passages.docid: 44440
negative_passages.text: Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 37219
negative_passages.text: Quadratic Integers in $\mathbb Q[\sqrt{-5}]$
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
negative_passages.docid: 36031
negative_passages.text: Finding modular of a fraction
negative_passages.docid: 105641
negative_passages.text: Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$