query: Prove that $2\sqrt{pq}$ is irrational, with $p$ and $q$ being primes and $p \neq q$
positive_passages.docid: 147777
positive_passages.text: If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
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: 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: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
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: 32848
negative_passages.text: Prove that the product of a non-zero rational and irrational number is irrational.
negative_passages.docid: 58284
negative_passages.text: How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?
negative_passages.docid: 87596
negative_passages.text: Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
negative_passages.docid: 5529
negative_passages.text: Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
negative_passages.docid: 111094
negative_passages.text: Help with proving that $\pi$ is irrational
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 158066
negative_passages.text: Prove $2^{1/3}$ is irrational.
negative_passages.docid: 58182
negative_passages.text: How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
negative_passages.docid: 89716
negative_passages.text: Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?
negative_passages.docid: 6275
negative_passages.text: How can you prove that the square root of two is irrational?