query: Prove: $2730|(n^{13}-n),n\in\mathbb N$
positive_passages.docid: 1895
positive_passages.text: How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
negative_passages.docid: 96903
negative_passages.text: How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
negative_passages.docid: 59462
negative_passages.text: Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
negative_passages.docid: 43940
negative_passages.text: Showing that $n$ is pseudoprime to the base $a$
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 141371
negative_passages.text: Prove that $\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$
negative_passages.docid: 65391
negative_passages.text: Prove by induction. $n \in \mathbb N$.
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 153123
negative_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 149549
negative_passages.text: Infimum of $n$-th root of $n$
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: 59612
negative_passages.text: Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
negative_passages.docid: 24403
negative_passages.text: Prove that $ n < 2^{n}$ for all natural numbers $n$.
negative_passages.docid: 27580
negative_passages.text: Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
negative_passages.docid: 67295
negative_passages.text: Please check my proof for $N$ is infinite
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 77579
negative_passages.text: Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$