query: {$a_n$} $\in N$ .if $(a_m,a_n)=(m,n)$ when $m \ne n$,prove that $a_n=n $ for all $n$
positive_passages.docid: 145397
positive_passages.text: How to prove that for every $i$, $a_i = i$
negative_passages.docid: 60804
negative_passages.text: About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
negative_passages.docid: 37346
negative_passages.text: If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
negative_passages.docid: 75966
negative_passages.text: Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 80916
negative_passages.text: How to prove that if $A\subseteq A\times A$, then $A=\varnothing$?
negative_passages.docid: 64654
negative_passages.text: Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
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: 80601
negative_passages.text: Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
negative_passages.docid: 89279
negative_passages.text: If $\sum a_n/k^n = 0$ for all $k$, then $a_n = 0$ for all $n$
negative_passages.docid: 111951
negative_passages.text: Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
negative_passages.docid: 98855
negative_passages.text: Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
negative_passages.docid: 98470
negative_passages.text: If $A \ne \varnothing$, then $\varnothing^A=\varnothing$
negative_passages.docid: 30134
negative_passages.text: For $a,n \in \Bbb N$, how to prove if $a^n-1$ is prime then $a=2$ and $n$ is prime?
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$