Geodesic
A class of primality criteria formulated in terms of the divisibility of binomial coefficients
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 167-183

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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. 
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     author = {Yu. V. Matiyasevich},
     title = {A class of primality criteria formulated in terms of the divisibility of binomial coefficients},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--183},
     publisher = {mathdoc},
     volume = {67},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a8/}
}
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Yu. V. Matiyasevich. A class of primality criteria formulated in terms of the divisibility of binomial coefficients. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 167-183. http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a8/