query: Between $n$ and $2n$ there is always a prime number.
positive_passages.docid: 149762
positive_passages.text: Are there any Combinatoric proofs of Bertrand's postulate?
negative_passages.docid: 43044
negative_passages.text: All natural numbers are equal.
negative_passages.docid: 119486
negative_passages.text: Upper and Lower bounds on the nth prime number
negative_passages.docid: 75733
negative_passages.text: Divisibility of prime numbers
negative_passages.docid: 22023
negative_passages.text: Infinitely number of primes in the form $4n+1$ proof
negative_passages.docid: 33786
negative_passages.text: Proof that a certain fraction is always an integer
negative_passages.docid: 17919
negative_passages.text: For every prime of the form $2^{4n}+1$, 7 is a primitive root.
negative_passages.docid: 34680
negative_passages.text: Percentage of primes among the natural numbers
negative_passages.docid: 37315
negative_passages.text: not both $2^n-1,2^n+1$ can be prime.
negative_passages.docid: 25428
negative_passages.text: Prime numbers of the form $ n^{34} -9$
negative_passages.docid: 143772
negative_passages.text: How can I algebraically prove that $2^n - 1$ is not always prime?
negative_passages.docid: 5692
negative_passages.text: Proving that there is a perfect square between $n$ and $2n$
negative_passages.docid: 75378
negative_passages.text: how to prove this extended prime number theorem?
negative_passages.docid: 56229
negative_passages.text: On the difference between consecutive primes
negative_passages.docid: 144592
negative_passages.text: how to prove the existence and uniqueness of the prime factorization of any natural number.
negative_passages.docid: 17233
negative_passages.text: are two consecutive numbers relatively prime?
negative_passages.docid: 83643
negative_passages.text: Are there prime gaps of every size?