query: Showing that if $p$ and $p^2+8$ are primes then $p^3+4$ is also prime
positive_passages.docid: 19979
positive_passages.text: Why does $p^2+8$ prime imply $p^3+4$ prime
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: 58114
negative_passages.text: For which primes p is $p^2 + 2$ also prime?
negative_passages.docid: 13478
negative_passages.text: Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
negative_passages.docid: 12365
negative_passages.text: Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 124995
negative_passages.text: $p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
negative_passages.docid: 148713
negative_passages.text: If $p\geq 5$ is a prime number, show that $p^2+2$ is composite.
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 72666
negative_passages.text: Prime Numbers And Perfect Squares
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 106111
negative_passages.text: Understanding that there are infinitely many primes of form $4n+3$
negative_passages.docid: 46803
negative_passages.text: Proof that there are infinitely many primes congruent to 3 modulo 4
negative_passages.docid: 38748
negative_passages.text: There are infinitely many prime numbers $p$ for which $p + 2$ and $p + 4$ are also prime numbers
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: 34906
negative_passages.text: Showing $[\mathbb{Q}(\sqrt[4]{2},\sqrt{3}):\mathbb{Q}]=8$.