Prove that $\sum_{i=1}^k i!$ can never be of the form $n^m$ (where $n,m\in\mathbb N$ and $m\ge 2$) for $k\ge 4$.
56624
Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
104447
Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
68308
A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
65672
Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
114575
If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
81124
Infinite product problem: $\sum p_n0$
120976
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
114810
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
104753
$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?
131771
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
52113
Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
146241
If $a \in \Bbb Z$ is the sum of two squares then $a$ can't be written in which of the following forms?
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
34666
Prove that the expression cannot be a power of 2
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.