If $p$ divides $a^n$, how to prove/disprove that $p^n$ divides $a^n$?
75733
Divisibility of prime numbers
122386
How do I prove that $3$ divides $14^{2n}-1$ by induction?
21781
Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
75373
Show that $2^n>n$ for $n\in \mathbb{N}$
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
113458
How to factor $a^n - b^n$?
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
100892
Proving divisibility of $a^3 - a$ by $6$
8494
Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.
144622
Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
140405
Show that there does not exist any $n$ for which $a^n=a$ forall $a\in \Bbb Z_m$ where $m$ is divisible by the square of some prime.
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
127293
Prove that $(a-b) \mid (a^n-b^n)$
152299
Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
34491
Is there a way to show that $\sqrt{p_{n}} < n$?
112147
Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$