# Gaussian Integers

## Tuesday, 25 December 2007

### Euler’s Prime Generating Equation

In the eighteenth century, Leonhard Euler discovered (no one knows how) that the equation

produced primes consecutively when fed with numbers n=0 to 39. Even though the equation fails when n=40 it still manages to spit out more prime numbers than any other quadratic equation. Of the first 10 million values the proportion of primes is about one in three. Euler then showed that you can substitute 41 with k=2,3,5,11,17 and the formula

would still produce primes from n=0 to k-2.