Let $p \ge 5$ be prime. Prove $p^2 \equiv 1 \pmod{24}$
15915
For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
100696
Let $p$ be prime. Consider the sequence: $1, 2^2, 3^3, 4^4, 5^5, 6^6 \cdots \pmod p$. Prove the sequence is periodic with smallest period $p(p-1).$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
40156
Determine all primes $p$ for which $5$ is a quadratic residue modulo $p$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
124241
Finding all integers $k \geq 2$ such that $k^2 \equiv 5k \pmod{15}$. What is going on here?
77430
Prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$
54286
Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
118574
prove that $24 \mid a(a^2-1)$
126893
A group of order 2p (p prime) and other conditions - prove abelian.
32803
Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
152591
For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?
147101
Show that if $p>5$ is prime, then $240 \mid (p^4-1)$