$\binom{p^{\alpha}-1}{k} = (-1)^k\pmod{p}$?
18634
Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
62382
I suppose $(a,m)=(a-1,m)=1$, show that $1+a+a^2+\ldots+a^{\phi(m)-1}\equiv0\pmod m$
75910
Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
143365
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
151961
Irreducibility of $x^{p(p-1)}+x^{p(p-2)}+\cdots+x^p+1$ over $\mathbb Q$
150822
Finding $\sum_{k=0}^{n-1}\frac{\alpha_k}{2-\alpha_k}$, where $\alpha_k$ are the $n$-th roots of unity
14399
Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$
81729
Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
118571
Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
97196
$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
120554
highest power of prime $p$ dividing $\binom{m+n}{n}$
21874
Solve $x^3+x \equiv 1 \pmod p$
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$