query: For which natural numbers $n$ is $\sqrt n$ irrational? How would you prove your answer?
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: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 88429
negative_passages.text: Prove $\sqrt{2} + \sqrt{5}$ is irrational
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 120906
negative_passages.text: Is there a proof that $\pi \times e$ is irrational?
negative_passages.docid: 28347
negative_passages.text: Prove that $\sqrt 2 + \sqrt 3$ is irrational
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 92226
negative_passages.text: Prove $\sqrt6$ is irrational
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 75204
negative_passages.text: Irrationality $\sqrt{2}$ with ideal
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 109175
negative_passages.text: Why is it hard to prove whether $\pi+e$ is an irrational number?
negative_passages.docid: 143733
negative_passages.text: Is this proof that $\sqrt 2$ is irrational correct?
negative_passages.docid: 81428
negative_passages.text: Why does proof by contradiction not work on Rational numbers but only on Irrational numbers?
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
negative_passages.docid: 10008
negative_passages.text: Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers
negative_passages.docid: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational