Miller–Rabin primality test

The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test. Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven Extended Riemann hypothesis;Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

Just like the Fermat and Solovay–Strassen tests, the Miller–Rabin test relies on an equality or set of equalities that hold true for prime values, then checks whether or not they hold for a number that we want to test for primality.

First, a lemma about square roots of unity in the finite field Z/pZ, where p is prime and p > 2. Certainly 1 and −1 always yield 1 when squared modulo p; call these trivial square roots of 1. There are no nontrivial square roots of 1 modulo p (a special case of the result that, in a field, a polynomial has no more zeroes than its degree). To show this, suppose that x is a square root of 1 modulo p. Then:

In other words, prime p divides the product (x − 1)(x + 1). By Euclid's lemma it divides one of the factors x − 1 or x + 1, implying that x is congruent to either 1 or −1 modulo p.

...
Wikipedia