query: Prove that the square root of 3 is irrational
positive_passages.docid: 1383
positive_passages.text: How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
negative_passages.docid: 153043
negative_passages.text: Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem
negative_passages.docid: 21457
negative_passages.text: Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 132115
negative_passages.text: Verify my proof of the irrationality of $\sqrt{6}$.
negative_passages.docid: 80304
negative_passages.text: Roots of $\,ax^2+bx+c\,$ are irrational if $a,b,c$ are odd integers
negative_passages.docid: 18360
negative_passages.text: Proof that the irrational numbers are uncountable
negative_passages.docid: 124641
negative_passages.text: Prove that $5^{1/3}+7^{1/2}$ is irrational
negative_passages.docid: 151153
negative_passages.text: Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
negative_passages.docid: 89716
negative_passages.text: Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?
negative_passages.docid: 88429
negative_passages.text: Prove $\sqrt{2} + \sqrt{5}$ is irrational
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 55077
negative_passages.text: prove that $2\sqrt5 +\sqrt{11}$ is irrational
negative_passages.docid: 5529
negative_passages.text: Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
negative_passages.docid: 44440
negative_passages.text: Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational
negative_passages.docid: 84558
negative_passages.text: Prove that the square root of a positive integer is either an integer or irrational