query: Prime $p \geq 5$ is of the form $6k + 1$ or of the form $6k + 5$
positive_passages.docid: 2425
positive_passages.text: Is that true that all the prime numbers are of the form $6m \pm 1$?
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: 44063
negative_passages.text: Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.
negative_passages.docid: 116361
negative_passages.text: Relationship between prime factorizations of $n$ and $n+1$?
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
negative_passages.docid: 105444
negative_passages.text: Possible values of prime gaps
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
negative_passages.docid: 15210
negative_passages.text: Group of order 6 proof
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: 82482
negative_passages.text: Proving an infinite number of primes of the form 6n+1
negative_passages.docid: 140412
negative_passages.text: If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
negative_passages.docid: 68221
negative_passages.text: Question regardles primes and the fundamental theorem of arithmetic
negative_passages.docid: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 43567
negative_passages.text: Some hints for "If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$"
negative_passages.docid: 56229
negative_passages.text: On the difference between consecutive primes
negative_passages.docid: 78342
negative_passages.text: When is $\phi(n)$ a prime number?