query: Find all natural numbers $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is rational...
positive_passages.docid: 142008
positive_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 108691
negative_passages.text: Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.
negative_passages.docid: 150576
negative_passages.text: When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$?
negative_passages.docid: 41486
negative_passages.text: If $x^n + a_{n-1}x^{n-1} + ... + a_0 = 0$ has a rational root, then such root is an integer
negative_passages.docid: 52951
negative_passages.text: Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
negative_passages.docid: 83752
negative_passages.text: If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 41641
negative_passages.text: $(n-7)(n+7)=$ some perfect square
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
negative_passages.docid: 44790
negative_passages.text: Find $n$ for which $\frac{n(n+1)}{2}$ is perfect square
negative_passages.docid: 16660
negative_passages.text: Prove that any positive number has an $n$th root
negative_passages.docid: 105225
negative_passages.text: Number Theory: Complete set of residues modulo $n$
negative_passages.docid: 127955
negative_passages.text: Show that $\sqrt{1+n(n+1)(n+2)(n+3)}$ is a whole number for all whole numbers $n$.
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 19588
negative_passages.text: $n!+1$ being a perfect square
negative_passages.docid: 12967
negative_passages.text: Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$