Lets us denote by $\mathbb{P}$ the set of prime numbers.
It is well known that, given an integer $p>1$ :
$$\boxed{p\in\mathbb{P}\Leftrightarrow\forall k\in\{1,\cdots,p-1\},\,p\mid\binom pk}$$
I have three questions :
1°) Does anyone know if this property has an official name ? Or by whom it has been discovered ?
2°) I do know that the implication $\Rightarrow$ can be used to provide a proof of the little Fermat theorem (by induction). Does anyone know of some other significative application of it ?
3°) The implication $\Leftarrow$ is certainly a very bad primality test ! But I guess that it could nevertheless have interesting applications. Any suggestion would be appreciated :)
This is a special case of Lucas' theorem, which more generally computes ${m \choose n} \bmod p$, and it is also a special case of Kummer's theorem, which more generally computes the largest power of $p$ that divides ${m \choose n}$. I don't know if this special case itself has a name.
(Or rather, $\Rightarrow$ straightforwardly follows from these results, and $\Leftarrow$ does too but maybe less straightforwardly.)
Probably the most famous application of $\Rightarrow$ is to prove that the Frobenius map $x \mapsto x^p$ is a ring homomorphism in characteristic $p$. I'm not aware of any applications of $\Leftarrow$ off the top of my head.