Prove that if $n^x$ is an integer for every natural number $n$, then $x$ is also an integer.
981
If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?
30136
Proof that a Combination is an integer
74713
Show that for any natural numbers, if $S(x)=S(y)$ then $x=y$.
121107
Proof by induction: $n^{n+1} > (n + 1)^n$, $(1 + x)^n \ge 1 + nx$, other inequalities
121874
Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
16235
Prove by induction that every positive integer is represented by some binary number?
155757
Number theory: $x^y + 1 = y^x$
75373
Show that $2^n>n$ for $n\in \mathbb{N}$
55266
Prove that $(n+1)^{n-1}<n^n$
68400
Proof that $\frac{(2n)!}{2^n}$ is integer
36906
Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
149716
Prove that $ \sqrt{n(n+1)(n+2)(n+3)+1} $ is always a natural number
22073
Prove by induction that $n^2<n!$
22367
How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
63183
Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
145519
Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$