query: Proof: For every positive integer $n$, there is a sequence of $n$ consecutive positive integers containing no primes.
positive_passages.docid: 22367
positive_passages.text: How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
negative_passages.docid: 112182
negative_passages.text: Sequence of positive integers such that their reciprocals are in arithmetic progression
negative_passages.docid: 3632
negative_passages.text: Basic number theory proofs
negative_passages.docid: 131476
negative_passages.text: consecutive prime power
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: 23544
negative_passages.text: Complete induction proof that every $n > 1$ can be written as a product of primes
negative_passages.docid: 16235
negative_passages.text: Prove by induction that every positive integer is represented by some binary number?
negative_passages.docid: 65978
negative_passages.text: How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two consecutive integers if $1$ and $n$ also count as consecutive?
negative_passages.docid: 47799
negative_passages.text: Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
negative_passages.docid: 24582
negative_passages.text: Is the set of sequences of natural numbers with trailing zeros countable?
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 33810
negative_passages.text: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
negative_passages.docid: 123117
negative_passages.text: Is it possible to find $n-1$ consecutive composite integers
negative_passages.docid: 81723
negative_passages.text: Infinitely many primes proof using Euler's totient
negative_passages.docid: 4061
negative_passages.text: Proving there are infinitely many pairs of square-full consecutive integers
negative_passages.docid: 59261
negative_passages.text: If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$.
negative_passages.docid: 156660
negative_passages.text: The set of all strictly increasing sequences of natural numbers