query: Let $p$ be a prime number for which also $p^2+2$ is a prime. Show that then $p^3+2$ is also a prime.
positive_passages.docid: 22742
positive_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 6700
negative_passages.text: An Olympiad problem in prime numbers :can we proof that "n" is prime in this case "
negative_passages.docid: 100984
negative_passages.text: Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
negative_passages.docid: 26503
negative_passages.text: Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
negative_passages.docid: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 7203
negative_passages.text: Prime numbers the rank of which is also a prime.
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: 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: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 1702
negative_passages.text: Proving that there are infinitely many primes with remainder of 2 when divided by 3
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: 72666
negative_passages.text: Prime Numbers And Perfect Squares
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 73515
negative_passages.text: Primes Number Theory
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: 10403
negative_passages.text: Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
negative_passages.docid: 86594
negative_passages.text: If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$