Prove that $\sum_{j = 0}^p {p \choose j}{p + j \choose j} \equiv 2^p + 1 ~(\text{mod}~p^2)$
54286
Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
156564
Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$
11693
If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
67752
Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
81124
Infinite product problem: $\sum p_n0$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
142132
How do you prove $\sum \frac {n}{2^n} = 2$?
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
122149
How to prove that $\sum_{i<j}(X_i-X_j)^2=n\sum_{i=1}^{n}(X_i-\bar{X})^2$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
23830
Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
7448
Prove $S \doteq \sum_{n=1}^{\infty} p_n  0$ assuming $0 \leq p_n < 1$.
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
148458
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$