query: Prove that $\sqrt{3}$ is not a rational number
positive_passages.docid: 17930
positive_passages.text: Please explain this step in proving the square root of 3 is irrational
negative_passages.docid: 37915
negative_passages.text: $\sqrt{2}$ is irrational proof using well-Ordering Property
negative_passages.docid: 21457
negative_passages.text: Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
negative_passages.docid: 82485
negative_passages.text: Show that the set of non zero rationals is not a cyclic group under multiplication.
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: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 37529
negative_passages.text: If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational
negative_passages.docid: 62058
negative_passages.text: How to prove $1$,$\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent over $\mathbb{Q}$?
negative_passages.docid: 83862
negative_passages.text: How to prove that the number $1!+2!+3!+...+n!$ is never square?
negative_passages.docid: 84558
negative_passages.text: Prove that the square root of a positive integer is either an integer or irrational
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 24601
negative_passages.text: Proving that a natural number made entirely of 6's and 0's is not a square.
negative_passages.docid: 33628
negative_passages.text: Prove that $\sin(\sqrt x)$ not periodic
negative_passages.docid: 139755
negative_passages.text: Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
negative_passages.docid: 123544
negative_passages.text: Show that $n = 3^{100} + 2$ is not a prime number.
negative_passages.docid: 92226
negative_passages.text: Prove $\sqrt6$ is irrational
negative_passages.docid: 87764
negative_passages.text: Prove that the difference between two rational numbers is rational