query: Prove that $\sqrt{2} +\sqrt{n}$ is irrational
positive_passages.docid: 22800
positive_passages.text: Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
negative_passages.docid: 122226
negative_passages.text: Incorrect logic in popular proof of the irrationality of $\sqrt2$?
negative_passages.docid: 64885
negative_passages.text: Irrational and rational sequence proof
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 92226
negative_passages.text: Prove $\sqrt6$ is 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: 50930
negative_passages.text: Proving the continued fraction representation of $\sqrt{2}$
negative_passages.docid: 40009
negative_passages.text: Show that $e^{\sqrt 2}$ is irrational
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
negative_passages.docid: 140949
negative_passages.text: Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational
negative_passages.docid: 62485
negative_passages.text: How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$
negative_passages.docid: 7966
negative_passages.text: How prove $ \sqrt{2}+\sqrt{3}>\pi$?
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 114793
negative_passages.text: Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational
negative_passages.docid: 131634
negative_passages.text: Prove the irrationality of $0.235711131719...$
negative_passages.docid: 32848
negative_passages.text: Prove that the product of a non-zero rational and irrational number is irrational.
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational