query: There always exists a sequence of consecutive composite integers of length $n$ for all $n$.
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: 1507
negative_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial
negative_passages.docid: 103581
negative_passages.text: Which positive integers can NOT be written as a sum of consecutive positive integers
negative_passages.docid: 45885
negative_passages.text: If there exist sequence such that $g(x_n)=f(x_{n+1})$, then we have $g(x_0)=f(x_0)$ for some $x_0$
negative_passages.docid: 73088
negative_passages.text: Give an example of a sequence of real numbers with subsequences converging to every real number
negative_passages.docid: 150385
negative_passages.text: The use of the Axiom of Choice in this proof of the sequence lemma
negative_passages.docid: 127855
negative_passages.text: Show that the product of any $m$ consecutive positive integers is divisible by $m!$
negative_passages.docid: 56985
negative_passages.text: Show that every positive integer has a unique representation in the form $50k+l$...?
negative_passages.docid: 48850
negative_passages.text: Consecutive composite numbers
negative_passages.docid: 124448
negative_passages.text: For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.
negative_passages.docid: 45788
negative_passages.text: $n$-th digit in the sequence of natural numbers
negative_passages.docid: 94575
negative_passages.text: Inductive proof that there are infinitely many pairs of consecutive powerful integers?
negative_passages.docid: 28099
negative_passages.text: Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers.
negative_passages.docid: 37330
negative_passages.text: Prove that any two consecutive terms of the Fibonacci sequence are relatively prime
negative_passages.docid: 144575
negative_passages.text: Factoring n, where n=pq and p and q are consecutive primes
negative_passages.docid: 59535
negative_passages.text: Prove that there exists such a set of $n$ positive integers
negative_passages.docid: 29065
negative_passages.text: Encode each $n_1,n_2,n_3,...∈N^N$ by an inﬁnite sequence of 0s and 1s with inﬁnitely many 0s, and give a proof that $N^N$ is equinumerous with $R$.