Formula for primes

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be.

A simple formula is

By Wilson's theorem, ${\displaystyle n+1}$ is prime if and only if ${\displaystyle n!{\bmod {(}}n+1)=n}$. Thus, when ${\displaystyle n+1}$ is prime, the first factor in the product becomes one, and the formula produces the prime number ${\displaystyle n+1}$. But when ${\displaystyle n+1}$ is not prime, the first factor becomes zero and the formula produces the prime number 2. This formula is not an efficient way to generate prime numbers because evaluating ${\displaystyle n!{\bmod {(}}n+1)}$ requires about ${\displaystyle n-1}$ multiplications and reductions ${\displaystyle {\bmod {(}}n+1)}$.

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