query: Proof that $\sqrt[n]{a+1}$ and $\sqrt[n]{a-1}$ cannot be both rationals
positive_passages.docid: 150088
positive_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 67031
negative_passages.text: use contradiction to prove that the square root of $p$ 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: 64885
negative_passages.text: Irrational and rational sequence proof
negative_passages.docid: 86893
negative_passages.text: Show that it is impossible to list the rational numbers in increasing order
negative_passages.docid: 28347
negative_passages.text: Prove that $\sqrt 2 + \sqrt 3$ is irrational
negative_passages.docid: 134885
negative_passages.text: proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $
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: 8589
negative_passages.text: $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ not an integer
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$
negative_passages.docid: 92703
negative_passages.text: Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
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: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
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: 151593
negative_passages.text: Prove that the limit of $\sqrt{n+1}-\sqrt{n}$ is zero
negative_passages.docid: 30511
negative_passages.text: Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]