Erastosthenes (276 - 196 B.C.)

Eratosthenes invented a method for efficiently constructing tables of prime numbers. This method, the "Sieve of Eratosthenes", It goes like this. First, write down a list of integers beginning with 2 and ending with some number, say, 20:

  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Then mark all multiples of 2:

  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
      x   x   x    x     x     x     x     x     x

Move to the next unmarked number, which in this case is 3, then mark all its multiples:

  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
      x   x   x x  x     x     x  x  x     x     x

Continue in this fashion, marking all multiples of the next unmarked number until there are no new unmarked numbers. The final result is in fact the last table given, so the primes not exceeding 20 are

  2 3 5 7 11 13 17 19