There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers.
Many of these alternate statements characterizing primality are related to Wilson's theorem, or are restatements of this classical result given in terms of other special variants of generalized factorial functions.
For instance, new variants of Wilson's theorem stated in terms of the hyperfactorials, subfactorials, and superfactorials are given in.
is odd, we have that Clement's congruence-based theorem characterizes the twin primes pairs of the form
through the following conditions: 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 The prime pairs of the form
We have elementary congruence-based characterizations of the primality of such pairs, proved for instance in the article.
given by 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.