query: Will there at some point be more numbers with $n$ factors than prime numbers for any $n$?
positive_passages.docid: 16459
positive_passages.text: Limit of ratios of numbers with $m$ factors and primes
negative_passages.docid: 37947
negative_passages.text: Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$.
negative_passages.docid: 53967
negative_passages.text: distribution of prime numbers
negative_passages.docid: 53525
negative_passages.text: Infiniteness of non-twin primes.
negative_passages.docid: 28228
negative_passages.text: Is a prime factor of a number always less than its square root?
negative_passages.docid: 132132
negative_passages.text: Are there infinitely many primes of the form $6^{2n}+1$ or only finitely many?
negative_passages.docid: 62061
negative_passages.text: Do we have negative prime numbers?
negative_passages.docid: 33505
negative_passages.text: Show that every large integer has a large prime-power factor
negative_passages.docid: 37315
negative_passages.text: not both $2^n-1,2^n+1$ can be prime.
negative_passages.docid: 144575
negative_passages.text: Factoring n, where n=pq and p and q are consecutive primes
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 126471
negative_passages.text: At least one prime between $\sqrt{n}$ and $n$?
negative_passages.docid: 52530
negative_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
negative_passages.docid: 134066
negative_passages.text: Total number of factors of a number
negative_passages.docid: 54128
negative_passages.text: Is it feasible to build an index of prime factors?
negative_passages.docid: 13739
negative_passages.text: For all $n>2$ there exists a prime number between $n$ and $ n!$