How to prove that $1^n+2^n+...+(p-1)^n \equiv 0\pmod p$?
152
What is $1^k+2^k+\cdots+ (p-1)^k$ modulo $p$? (From Ireland and Rosen).
37093
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
157294
Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
109532
Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
55266
Prove that $(n+1)^{n-1}<n^n$
99052
Prove that $( 1 + n^{-2}) ^n \to 1$.
89175
Proof of $(a+b)^{n+1}$
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
134348
When is $(p - 1)! + 1$ a power of $p$?
4315
show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
128107
find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
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 $a$.
148458
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
102234
Show $a^p \equiv a \mod 2p$