Let $p\geq 5$ be a prime. Show that $\sum_{k=0}^{\frac {p-1}{2}} {\binom p k}3^k \equiv 2^p-1 \pmod {p^2}$
132984
$\sum\limits_{k=0}^{\frac{p-1}2}3^k\binom{p}{k}\equiv 2^p-1\pmod{p^2}$
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
148458
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
123033
Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
21791
Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
38876
$r! \equiv (−1)^k \pmod p$
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
21629
Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
21874
Solve $x^3+x \equiv 1 \pmod p$
5841
Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
19581
$\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
147307
How to evaluate $\sum_{\gcd (p,q)=1} \frac{1}{p^2q^2}$?