A class of primality criteria formulated in terms of the divisibility of binomial coefficients

We find a class of theorems of the type "$q$ is a prime number iff $R(g)$ is a divisor of the binomial coefficient $\begin{pmatrix}S(q)\\T(q)\end{pmatrix}$"; here $R$, $S$, $T$ are certain fully significant functions that are superpositions of addition, subtraction, multiplication, division, and raising to a power. Similar criteria were also obtained for prime Mersenne numbers, prime Fermat numbers, and twin-prime numbers.