query: The n-th root of a prime number 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: 44600
negative_passages.text: How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
negative_passages.docid: 101162
negative_passages.text: Square root of $2$ is irrational
negative_passages.docid: 32
negative_passages.text: Is $n^{th}$ root of $2$ an irrational number?
negative_passages.docid: 30478
negative_passages.text: How to prove that a number is irrational
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 150300
negative_passages.text: Root of unity for $n+1$
negative_passages.docid: 44573
negative_passages.text: Can $\pi$ be rational in some base radix
negative_passages.docid: 30411
negative_passages.text: The closure of the irrational numbers on $\mathbb{R}$
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 48975
negative_passages.text: Transcendental numbers and irrationality measure
negative_passages.docid: 141424
negative_passages.text: $\log_7 n$ is either an integer or an irrational number
negative_passages.docid: 144244
negative_passages.text: Bound on difference between irrational square root and rational number
negative_passages.docid: 44440
negative_passages.text: Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational
negative_passages.docid: 114793
negative_passages.text: Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational
negative_passages.docid: 147777
negative_passages.text: If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?