**This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.**

### Abstract

The most efficient known primality tests are
*probabilistic* in the sense that they use
randomness and may, with some probability, mistakenly classify a
composite number as prime – but never a prime number as
composite. Examples of this are the Miller–Rabin test, the
Solovay–Strassen test, and (in most cases) Fermat's
test.

This entry defines these three tests and proves their correctness. It also develops some of the number-theoretic foundations, such as Carmichael numbers and the Jacobi symbol with an efficient executable algorithm to compute it.