If $n>$1 and $S_n= \frac12 + \frac13 +\cdots + \frac1n$, show $S_n$ is not an integer.
46946
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
97684
Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
99631
Prove/Disprove: Find $a\in S_4$ that is not a power of $b$ such that $ab=ba$ while $b=(123)$
32917
Show that, for $t>0$, $\log t$ is not a polynomial.
36133
Show that $\inf(\frac{1}{n})=0$.
53857
Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
93948
Is my proof correct? $p_1p_2p_3\cdots p_n+1)$ cannot be the square of an integer
156766
Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
109440
How does $(12\dots n)$ and $(a\, b)$ generate $S_n$?
63474
Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
75461
If $n>m$, then the number of $m$-cycles in $S_n$ is given by $\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}$.
51928
Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
41051
How to show $e^{e^{e^{79}}}$ is not an integer
51382
Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
24444
Prove that $371\cdots 1$ is not prime
114873
Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.