query: For every positive even number $n$, does there always exist a pair of prime $(p,q)$ such that $p-q =n$?
positive_passages.docid: 21383
positive_passages.text: Can every even integer be expressed as the difference of two primes?
negative_passages.docid: 132132
negative_passages.text: Are there infinitely many primes of the form $6^{2n}+1$ or only finitely many?
negative_passages.docid: 17233
negative_passages.text: are two consecutive numbers relatively prime?
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 13739
negative_passages.text: For all $n>2$ there exists a prime number between $n$ and $ n!$
negative_passages.docid: 53525
negative_passages.text: Infiniteness of non-twin primes.
negative_passages.docid: 131385
negative_passages.text: How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
negative_passages.docid: 75166
negative_passages.text: Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
negative_passages.docid: 28228
negative_passages.text: Is a prime factor of a number always less than its square root?
negative_passages.docid: 131551
negative_passages.text: $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
negative_passages.docid: 45454
negative_passages.text: Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
negative_passages.docid: 58182
negative_passages.text: How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
negative_passages.docid: 100426
negative_passages.text: Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
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: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 44857
negative_passages.text: Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$