query: How do you prove $\sqrt{n}$ is an integer or it 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: 121694
negative_passages.text: How to prove $\log n \leq \sqrt n$ over natural numbers?
negative_passages.docid: 75204
negative_passages.text: Irrationality $\sqrt{2}$ with ideal
negative_passages.docid: 78287
negative_passages.text: Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational.
negative_passages.docid: 31245
negative_passages.text: How to prove that Fibonacci number is integer?
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 99311
negative_passages.text: Prove $2^{1/3} + 2^{2/3}$ is irrational
negative_passages.docid: 120906
negative_passages.text: Is there a proof that $\pi \times e$ is irrational?
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 86988
negative_passages.text: Cardinality of the Irrationals
negative_passages.docid: 107105
negative_passages.text: Is an irrational number odd or even?
negative_passages.docid: 126074
negative_passages.text: Proof of the irrationality of $\sqrt{3}$ - logic question
negative_passages.docid: 143733
negative_passages.text: Is this proof that $\sqrt 2$ is irrational correct?
negative_passages.docid: 37915
negative_passages.text: $\sqrt{2}$ is irrational proof using well-Ordering Property
negative_passages.docid: 111094
negative_passages.text: Help with proving that $\pi$ is irrational
negative_passages.docid: 129433
negative_passages.text: $\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
negative_passages.docid: 96204
negative_passages.text: Irrational number $\sqrt2$ proof