The sieve of Eratosthenes and the tetractys
A new look at the oldest method for finding primes. The numbers that drop out of the twin positions between 25 and 169 arrange themselves in grids 10 apart.
The sieve of Eratosthenes is the oldest method for finding primes: write the numbers down and cross out, one after another, all multiples of 2, 3, 5, 7 … What remains are the primes.
But the sieve can also be read differently. Take the rhythm of six as given: all multiples of 2 and 3 are already crossed out, leaving the twin positions 6n ± 1. The interesting work then begins only at 25, where five disturbs the order of the twins for the first time.
The starting point: up to 24 every twin position holds a prime. With 25 and 35 the first ones drop out.
The crossed-out positions
Exactly 10 numbers after 25, at 35, a grid of 14 begins that sieves out the multiples of 7. Then at 45 comes a grid of 18, and so on. To see the order, include the odd numbers divisible by three that lie outside the twin positions.
| Divisor | Start | Step | numbers hit up to 169 |
|---|---|---|---|
| 5 | 25 = 5 × 5 | 10 | 2535455565758595105115125135145155165 |
| 7 | 35 = 5 × 7 | 14 | 3549637791105119133147161 |
| 9 | 45 = 5 × 9 | 18 | 45638199117135153 |
| 11 | 55 = 5 × 11 | 22 | 557799121143165 |
| 13 | 65 = 5 × 13 | 26 | 6591117143169 |
| 15 | 75 = 5 × 15 | 30 | 75105135165 |
| 17 | 85 = 5 × 17 | 34 | 85119153 |
| 19 | 95 = 5 × 19 | 38 | 95133 |
| 21 | 105 = 5 × 21 | 42 | 105147 |
All twin positions between 25 and 169 that hold no prime (red), and the odd numbers divisible by three (green) lying between them as axes of symmetry.
The numbers divisible by three form the ordering axes of symmetry between the individual divisibility figures of the fanned-out simplexes. The whole table is mirror-symmetric too: the numbers in the grids repeat in a mirror-image half, because, for example, 5 × 7 and 7 × 5 hit the same number.
This order, grids offset by 10 each time with intervals growing by 4, continues without end. It recalls once more the beginning of the tetractys: 1 + 2 + 3 + 4 = 10.
Precisely these grids of numbers divisible by three probably also inspired Felix Stoffel’s sieve of 30. On this basis Stoffel divided the primes into “four temperaments”, which is also a view of the tetractys.