query: Prove there exist a prime $p$ between $n^3$ and $(n+1)^3-1$
positive_passages.docid: 26878
positive_passages.text: Primes between consecutive cubes
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: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
negative_passages.docid: 126494
negative_passages.text: Prove that there doesn't exist any integer $x \ge 3$ such that $x^2-1$ is prime.
negative_passages.docid: 158106
negative_passages.text: Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
negative_passages.docid: 13739
negative_passages.text: For all $n>2$ there exists a prime number between $n$ and $ n!$
negative_passages.docid: 52530
negative_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 144622
negative_passages.text: Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
negative_passages.docid: 51373
negative_passages.text: $G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 39915
negative_passages.text: Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
negative_passages.docid: 146512
negative_passages.text: Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
negative_passages.docid: 84603
negative_passages.text: Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
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: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 8710
negative_passages.text: Bounds for $n$-th prime