THE SIEVE OF ERATOSTHENESAn ancient method for finding the

prime numbers

ERA-WHO?

Eratosthenes was a Greek mathematician from Cyrene (modern day Libya!). He was born in 276 BC, and lived until about 195 BC.

ERA-WHAT? Eratosthenes was a remarkable

mathematician, he is credited with being the first to calculate the circumference of the earth as well as the tilt of its axis.

He may have calculated the distance between the earth to the sun, and the idea of “leap day”.

He is the father of the discipline of geography, making the first world map which utilized parallels and meridians.

He was the third “librarian” at Alexandria. He developed his famous sieve, an efficient

method of determining “small” prime numbers (those less than a million).

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

WHAT’S THE SECRET OF THE SIEVE?

Start with a list of numbers, where you want to know which are prime, and which are composite. For this illustration, we will use only numbers up through 100.

Cross out the number 1: it is neither prime nor composite.

Start with the next number, 2, and cross out all of its multiples. You will soon see that they make a pattern in the grid….

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

WHAT’S THE SECRET OF THE SIEVE?

Continue with the next number, 3, and cross out all of ITS multiples. They also form a pattern in the grid.

The next available number is 5. (Why is 4 not available?) Do the same thing here with it’s multiples….

Let’s cross out the multiples of 7 next…

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

WHAT’S THE SECRET OF THE SIEVE?

Continuing the process with 11, gives no new numbers crossed out, but if our grid included larger numbers, it would. The same is true for all remaining values…..

What remains are all the prime numbers less than 100