Abstract
We find a class of theorems of the type “q is a prime number iff R(q) is a divisor of the binomial coefficient\(\left( {\begin{array}{*{20}c} {S(q)} \\ {T(q)} \\ \end{array} } \right)\) ”; 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.
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