query: Prove that $\sqrt{p}$ is irrational where $p$ is prime
positive_passages.docid: 2926
positive_passages.text: The contradiction method used to prove that the square root of a prime is irrational
negative_passages.docid: 42376
negative_passages.text: Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$
negative_passages.docid: 26109
negative_passages.text: Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
negative_passages.docid: 87596
negative_passages.text: Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
negative_passages.docid: 109175
negative_passages.text: Why is it hard to prove whether $\pi+e$ is an irrational number?
negative_passages.docid: 50596
negative_passages.text: $p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 158161
negative_passages.text: $\mathbb{Q}(\sqrt p_{1}+\sqrt p_{2})=\mathbb{Q}(\sqrt p_{1},\sqrt p_{2})$ for $p_{1},p_{2}$ primes.
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 93898
negative_passages.text: Prove that for any prime p, there is a complete set of mod p residues {q1, . . . , qp}, where each qi is prime.
negative_passages.docid: 6275
negative_passages.text: How can you prove that the square root of two is irrational?
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 143733
negative_passages.text: Is this proof that $\sqrt 2$ is irrational correct?
negative_passages.docid: 37915
negative_passages.text: $\sqrt{2}$ is irrational proof using well-Ordering Property
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
negative_passages.docid: 21237
negative_passages.text: $\sqrt{6}-\sqrt{2}-\sqrt{3}$ is irrational