We learned that $(p-1)!\congruent -1 \mod p$ (Wilson's theorem). Which means that $p|(p-1)!+1$. A prime number which has the property that $p^2|(p-1)!+1$ is known as a **Wilson prime**. One example of a Wilson prime is 5, because $(4-1)!\congruent -1 \mod 5^2$. Currently only 3 Wilson primes are known (5, 13, and 563), however it is conjectured that there are an infinite amount of them. It is also conjectured that between any number x and y there exists $\log (\frac{\log y}{\log x})$ many Wilson primes.

There are also numbers known as **Wilson composites**, which is defined similarly.