query: $\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$
positive_passages.docid: 18634
positive_passages.text: Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 118674
negative_passages.text: Simple upper bound for $\binom{n}{k}$
negative_passages.docid: 147079
negative_passages.text: $\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
negative_passages.docid: 72026
negative_passages.text: Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
negative_passages.docid: 50413
negative_passages.text: Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
negative_passages.docid: 108401
negative_passages.text: Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
negative_passages.docid: 79940
negative_passages.text: $\binom{p}{i}$ divisible by $p$, with $p$ prime
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
negative_passages.docid: 132587
negative_passages.text: Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 13525
negative_passages.text: Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
negative_passages.docid: 101057
negative_passages.text: Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
negative_passages.docid: 22962
negative_passages.text: How to show $\binom{2p}{p} \equiv 2\pmod p$?
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
negative_passages.docid: 9522
negative_passages.text: Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$