# Sieve of Eratosthenes

The **sieve of Eratosthenes** is a simple algorithm for finding all the prime numbers up to a specified integer.

**Step 1.** List the integers, starting with "2".

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

**Step 2.** Mark the first number in the list as prime.

- Known primes: 2

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

**Step 3.** Step through the main list eliminating all multiples of the number just added to the list of known primes.

- Known primes: 2

- Main list: 3 5 7 9 11 13 15 17 19

**Step 4.** If the largest number in the main list is less than the square of the largest number in the known prime list, mark all numbers in the main list as prime; otherwise, return to Step 2.

- Since 19 is greater than the square of 2 (4), we return to Step 2:

- Known primes: 2 3

- Main list: 5 7 9 11 13 15 17 19

- Then step 3:

- Known primes: 2 3

- Main list: 5 7 11 13 17 19

- 19 is greater than the square of 3 (9), so we return to step 2:

- Known primes: 2 3 5

- Main list: 7 11 13 17 19

- Then step 3 (no changes to either list).

- 19 is less than the square of 5 (25), so the remaining list is prime.

- RESULT: The primes in the range 2 to 20 are: 2, 3, 5, 7, 11, 13, 17, 19.

## Reference

Κοσκινον Ερατοσθενους *or, The Sieve of Eratosthenes. Being an Account of His Method of Finding All the Prime Numbers*, by the Rev. Samuel Horsley, F. R. S.,
Philosophical Transactions (1683-1775), Vol. 62. (1772), pp. 327-347.