query: Prove that for $n>2$, $n$ cannot be a factor of $n+1$
positive_passages.docid: 17233
positive_passages.text: are two consecutive numbers relatively prime?
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 959
negative_passages.text: Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 131385
negative_passages.text: How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
negative_passages.docid: 42526
negative_passages.text: The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 24403
negative_passages.text: Prove that $ n < 2^{n}$ for all natural numbers $n$.
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 81267
negative_passages.text: A simple doubt in number theory problem.
negative_passages.docid: 55752
negative_passages.text: Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
negative_passages.docid: 107725
negative_passages.text: Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
negative_passages.docid: 77816
negative_passages.text: Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 54606
negative_passages.text: Number of factors of a big number
negative_passages.docid: 89381
negative_passages.text: Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.