The Sieve of Eratosthenes
Created | Updated Mar 7, 2003
Eratosthenes of Cyrene was a 3rd century bce Greek mathematician and astronomer. He was a director at the Library at Alexandria. He was the first mathematician to accurately estimate the diameter of Earth. He gave his name to The Sieve of Eratosthenes.
The sieve is a device designed to identify prime numbers, the ones divisible only by themselves and one. They begin 2, 3, 5, 7, 11, etc. The sieve takes different shapes depending on how many columns are used to arrange the numbers.
Eratosthenes used a natural factoring sieve to illustrate what prime numbers are. This sieve is built like a pyramid on a piece of graph paper, with the base being an infinite row of ones. Every continuous box in the row is shaded.
The next row is built on top of the first. Every second box is shaded, into infinity.
The next row is built on top of the second. Every third box is shaded.
The next row is the first composite row. Every fourth box is shaded forever. In every fourth column, there will already be a shaded 2.
As you continue to build upwards and across to shade the multiples of each new number, you are creating a natural factoring table. If a new row across begins with a prime number, that column will be empty of factors. This sieve functions well as a teaching device because you can see why a number is prime.
If you change the number of columns you arrange numbers in, the numbers line up in special ways. Every number of columns produces patterns, but only one is perfect. That is the sieve built upon a grid of base 6.
| 01 | 02 | 03 | 04 | 05 | 06 |
| 07 | 08 | 09 | 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 |
This table illustrates my point.
Columns 2,3,4 and 6 erase all multiples of 2 and 3 forever. Therefore, prime numbers will only be found in columns 1 and 5.
Certain multiples of primes infect the possible prime columns. In order to identify every prime number, simply cross off the multiple of primes. They were nice enough to remain orderly.
There is an imaginary position above the 6th column. This is MONAD. MONAD sets the size of the unit and the extent of the base line. That's why it hovers above the 6th column, because this is a base 6 grid.
Begin with the first prime not already sieved out, the multiples of five.
Begin at MONAD and go down one square, and left one square to 5,
down one left one, 10,
down one left one, 15,
down one left one, 20,
down one left one, 25,
around, 30,
down one left one, 35, etc.
The grid is circular. It can be curved into a cylindar, and all of the patterns circle around and down.
Multiples of 11 create the pattern down two, left one.
Multiples of 17 create the pattern down three, left one.
On the other side of the sieve, in the 1's column, either remove multiples the hard way or the easy way. Both work. Either leave MONAD over the 6th column (the hard way), or move the MONAD to the left of the 1.
Simply establish a pattern and it continues forever.
As you proceed dwonward, numbers in the possible prime columns that haven't been crossed off by previous multiples, are prime.
Notice that every prime starting with 5 is plus or minus 1 away from a multiple of 6. Forever.
The sieve may be continued upward, into negative numbers. Leave MONAD above the 6 and proceed to the left with -1, -2, etc.
