query: If p and $8p^2+1$ are prime numbers then $p=3$?
positive_passages.docid: 157884
positive_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
negative_passages.docid: 123296
negative_passages.text: Determine whether a number is prime
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
negative_passages.docid: 128575
negative_passages.text: Elementary proof that $3$ is a primitive root of a Fermat prime?
negative_passages.docid: 144851
negative_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
negative_passages.docid: 107502
negative_passages.text: Let $p \geq 2$. Prove that if $2^p-1$ is prime then $p$ must also be prime.
negative_passages.docid: 11564
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
negative_passages.docid: 127993
negative_passages.text: Let $p$ be a prime. Suppose that $\gcd(a, b) = p$. Find $\gcd(a^2,b)$ for all integers $a$ and $b$.
negative_passages.docid: 15915
negative_passages.text: For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
negative_passages.docid: 19016
negative_passages.text: Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple
negative_passages.docid: 103360
negative_passages.text: Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
negative_passages.docid: 157766
negative_passages.text: Are primes (ignoring $2$) equally likely to be $1~\text{or}~3\pmod 4$?
negative_passages.docid: 27170
negative_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 130846
negative_passages.text: Prove or disprove: If $a \in \mathbb{Z}$, $a > 0$, and $p$ is a prime such that $p^4 \mid a^3$, then $p^2 \mid a$.