In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.

A simple formula is

for positive integer ${\displaystyle n}$, where ${\displaystyle \lfloor \ \rfloor }$ is the floor function, which rounds down to the nearest integer. By Wilson's theorem, ${\displaystyle n+1}$ is prime if and only if ${\displaystyle n!\equiv n\!\!\!{\pmod {n+1}}}$. 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 modulo ${\displaystyle n+1}$.

In 1964, Willans gave the formula

for the ${\displaystyle n}$th prime number ${\displaystyle p_{n}}$. This formula reduces to

that is, it tautologically defines ${\displaystyle p_{n}}$ as the smallest integer ${\displaystyle m}$ for which the prime-counting function ${\displaystyle \pi (m)}$ is at least ${\displaystyle n}$. This formula is also not efficient. In addition to the appearance of ${\displaystyle (j-1)!}$, it computes ${\displaystyle p_{n}}$ by adding up ${\displaystyle p_{n}}$ copies of ${\displaystyle 1}$; for example,

The articles What is an Answer? by Herbert Wilf (1982) and Formulas for Primes by Underwood Dudley (1983) have further discussion about the worthlessness of such formulas.

A shorter formula based on Wilson's theorem was given by J. P. Jones in 1975, using ${\displaystyle \mathrm {mod} }$ as a function: