query: If n isn’t mth power of any natural number then prove $n^{\frac{1}{m}}$ 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: 40009
negative_passages.text: Show that $e^{\sqrt 2}$ is irrational
negative_passages.docid: 86497
negative_passages.text: Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$
negative_passages.docid: 27494
negative_passages.text: $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$
negative_passages.docid: 126074
negative_passages.text: Proof of the irrationality of $\sqrt{3}$ - logic question
negative_passages.docid: 109175
negative_passages.text: Why is it hard to prove whether $\pi+e$ is an irrational number?
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 39021
negative_passages.text: Proving that $\sqrt{2}+\sqrt{3}$ is irrational
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: 42444
negative_passages.text: Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 144836
negative_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
negative_passages.docid: 109953
negative_passages.text: How to prove that $\sqrt 3$ is an irrational number?
negative_passages.docid: 124641
negative_passages.text: Prove that $5^{1/3}+7^{1/2}$ is irrational
negative_passages.docid: 158066
negative_passages.text: Prove $2^{1/3}$ is irrational.
negative_passages.docid: 32
negative_passages.text: Is $n^{th}$ root of $2$ an irrational number?