Let $P$ be a $n\times n$ matrix, if there is $k\in \mathbb Z^+$ such that $P^k=O$, prove that $P^n=O$.
41411
How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
1258
Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$
55031
Show any matrix $A_{n\times n}$ can be written as sum of two nonsingular matrices
44328
Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
70861
$\mathbb P^n\times \mathbb P^m\not\cong \mathbb P^{m+n}$
133017
Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
65218
Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
50927
If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
114277
Proving in Discrete Structures Problem that $2^N = \binom{N}{0} + \binom{N}{1} + \binom{N}{2} + \dots + \binom{N}{N}$
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
92583
Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
145980
For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?
30456
The number of $2 × 2$ invertible matrices with entries from field $\mathbb{Z}_p$
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
51258
Matrix raised to a matrix: $M^N$, is this possible? with $M,N\in M_n(\Bbb K).$