Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
44063
Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.
21781
Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
36943
How to prove that $L=${$a^p$: p is prime} isn't regular?
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
66861
Show that $11^{10} \equiv 1 (\mod 100)$
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
64782
How to prove if $n$ is prime and $n | a^2$ then $n | a$?
42780
Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
80144
Prove that $x-1$ divides $x^n-1$
37947
Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$.
71788
Prove existence of multiplicative inverse modulo p when p is prime
20980
Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$