query: Show that for every $n > 1$ there exist $n$ consecutive composite numbers
positive_passages.docid: 142956
positive_passages.text: Prove that for any positive integer n there is a sequence of n consecutive composite numbers.
negative_passages.docid: 123117
negative_passages.text: Is it possible to find $n-1$ consecutive composite integers
negative_passages.docid: 136885
negative_passages.text: Series of natural numbers which has all same digits
negative_passages.docid: 37330
negative_passages.text: Prove that any two consecutive terms of the Fibonacci sequence are relatively prime
negative_passages.docid: 105985
negative_passages.text: Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers
negative_passages.docid: 56229
negative_passages.text: On the difference between consecutive primes
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 34452
negative_passages.text: In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive?
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
negative_passages.docid: 62120
negative_passages.text: Limit of the ratio of consecutive Fibonacci numbers
negative_passages.docid: 155575
negative_passages.text: Proving a polynomial $f(x)$ composite for infinitely many $x$
negative_passages.docid: 133017
negative_passages.text: Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
negative_passages.docid: 111316
negative_passages.text: PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
negative_passages.docid: 41571
negative_passages.text: Prove that in a sequence of numbers $49 , 4489 , 444889 , 44448889\ldots$
negative_passages.docid: 144738
negative_passages.text: Prime factorizations of consecutive numbers, can they both contains powers of greater than or equal to $1000$?
negative_passages.docid: 73206
negative_passages.text: recursion-consecutive numbers
negative_passages.docid: 58103
negative_passages.text: number of permutations in which no two consecutive numbers are adjacent