Determine all positive integers $(a,n)$ such that $(a-1)!+1=a^n$
8884
Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
106030
Find all the positive integer
126354
Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
139383
Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
46741
Prove that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer without using concepts of abstract algebra
2197
Induction principle for $n!<n^n$
69921
Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
20438
Prove that every positive integer $n$ has a unique expression of the form: $2^{r}m$ where $r\ge 0$ and $m$ is an odd positive integer
28861
Given an invertible matrix $A$ such that all elements in $A$ and in $𝐴^{−1}$ are integers, find $|𝐴^4|$
90789
Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$
16660
Prove that any positive number has an $n$th root
97656
Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
100558
Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
142900
Find all positive integers a,b,c to the equation a!b!= a!+b!+c!
9920
$(n^2 - 1)!$ is a multiple of $(n!)^n$
27580
Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$