query: If $n^s\in \mathbb{N}$ for all $n\in\mathbb{N}$, must $s\in\mathbb{N}?$
positive_passages.docid: 981
positive_passages.text: If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?
negative_passages.docid: 40565
negative_passages.text: How can I prove that one of $n$, $n+2$, and $n+4$ must be divisible by three, for any $n\in\mathbb{N}$
negative_passages.docid: 157924
negative_passages.text: $\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 119524
negative_passages.text: For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$
negative_passages.docid: 72567
negative_passages.text: Does a set of $n$ independent vectors in $R^{n}$ always span $R^{n}$?
negative_passages.docid: 108696
negative_passages.text: If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence
negative_passages.docid: 144732
negative_passages.text: If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
negative_passages.docid: 92290
negative_passages.text: For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
negative_passages.docid: 26912
negative_passages.text: Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
negative_passages.docid: 52951
negative_passages.text: Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
negative_passages.docid: 133406
negative_passages.text: $n^a$ integral for all integer $n$ implies $a$ integral
negative_passages.docid: 66272
negative_passages.text: Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?
negative_passages.docid: 62566
negative_passages.text: Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
negative_passages.docid: 39453
negative_passages.text: How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
negative_passages.docid: 112707
negative_passages.text: Must $Q^\mathsf{T}Q = I$ if $(Qx)^\mathsf{T}(Qx) = x^\mathsf{T}x$
negative_passages.docid: 135814
negative_passages.text: Prove certain subsets of $\mathbb R^n$ are subspaces