Table of congruences

For integers ${\displaystyle k\geq 1}$, we have the following form of Wilson's theorem:

${\displaystyle (k-1)!(p-k)!\equiv (-1)^{k}{\pmod {p}}\iff p{\text{ prime. }}}$

If ${\displaystyle p}$ is odd, we have that

${\displaystyle \left({\frac {p-1}{2}}\right)!^{2}\equiv (-1)^{(p+1)/2}{\pmod {p}}\iff p{\text{ an odd prime. }}}$

Clement's congruence-based theorem characterizes the twin primes pairs of the form ${\displaystyle (p,p+2)}$ through the following conditions:

${\displaystyle 4[(p-1)!+1]\equiv -p{\pmod {p(p+2)}}\iff p,p+2{\text{ are both prime. }}}$

P. A. Clement's original 1949 paper [2] provides a proof of this interesting elementary number theoretic criteria for twin primality based on Wilson's theorem. Another characterization given in Lin and Zhipeng's article provides that

${\displaystyle 2\left({\frac {p-1}{2}}\right)!^{2}+(-1)^{\frac {p-1}{2}}(5p+2)\equiv 0\iff p,p+2{\text{ are both prime. }}}$

The prime pairs of the form ${\displaystyle (p,p+2k)}$ for some ${\displaystyle k\geq 1}$ include the special cases of the cousin primes (when ${\displaystyle k=2}$) and the sexy primes (when ${\displaystyle k=3}$). We have elementary congruence-based characterizations of the primality of such pairs, proved for instance in the article.[3] Examples of congruences characterizing these prime pairs include

${\displaystyle 2k(2k)![(p-1)!+1]\equiv [1-(2k)!]p{\pmod {p(p+2k)}}\iff p,p+2k{\text{ are both prime, }}}$

and the alternate characterization when ${\displaystyle p}$ is odd such that ${\displaystyle p\not {\mid }(2k-1)!!^{2}}$ given by

${\displaystyle 2k(2k-1)!!^{2}\left({\frac {p-1}{2}}\right)!^{2}+(-1)^{\frac {p-1}{2}}\left[(2k-1)!!^{2}(p+2k)-(-4)^{k}\cdot p\right]\equiv 0\iff p,p+2k{\text{ are both prime. }}}$

Still other congruence-based characterizations of the primality of triples, and more general prime clusters (or prime tuples) exist and are typically proved starting from Wilson's theorem (see, for example, Section 3.3 in [4]).