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 (teal) 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.
The sieve in the plane
The sieve can also be laid out in two dimensions, perhaps as the Pythagoreans saw it: in the coordinate system of the division table, with numerators on one axis and denominators on the other. Numerator and denominator are equivalent and freely interchangeable. That is why the first cause of the fourfold lies in the even and odd numbers of both axes, and the resulting doubling necessarily shows in the quotient fields. Each prime lays a grid over the plane, wherever it divides numerator and denominator at once. Only the primes generate this matrix, all other numbers are already contained in it.
Numerators and denominators from 1 to 36. Coloured are the fields where one of the chosen primes divides both, lightly shaded the rows and columns of the primes. What stays white is coprime, the background of the one. As the limit grows, its share approaches 6/π² ≈ 61 %.
What we have here are figurate fractions, patterns like the ten-point triangle of the Pythagoreans. Their great advantage: they need no counting system. The decimal system is man-made and obscures the nature of number, for anyone who thinks in one counting system would immediately have to ask what holds in others. A dot pattern need not be counted in steps of ten, it can be grasped as by someone to whom numbers are foreign, a small child or a dolphin. The sieve of Eratosthenes, too, works independently of any counting system, and in the end the whole matrix dissolves into it. All coprime fields and the primes themselves are the remainder of the background, the one. This shows why for the Pythagoreans one was not really a number but the primal ground.
Looking at the first grids, 2 and 3 combine into a perfectly symmetrical pattern, the framework of the prime channels in the rhythm of six of the twins. Four goes into 2. So 1, 2, 3 and 4 secure the rhythm of six. From 5 on the channels become clogged again, and 5 × 5 = 25 is the first number to break the closed series of twins. Philosophically it is three that reconciles opposites, even and odd, active and passive, male and female, and joins them into the sixfold order. Philolaus already praised the harmonising power of three, the Freemasons still hold it sacred, and perhaps the Trinity and the creation in six days stand in this tradition too (see The order of the primes).
With the grid of 4 this perfect symmetry is complete. Tilt the table by 45° back into the lambdoma, and at its apex the ten-point triangle describes exactly this state of affairs. One meets it everywhere in the table where a new order begins and an old one ends, for the tetractys is a graded system. Above divisor 2, where the sieve first takes effect, the grids of struck-out numbers each begin ten further on, with gaps growing by 4:
| Grid | starts at |
|---|---|
| steps of 10 | 25 = 5 × 5 |
| steps of 14 | 35 = 5 × 7 |
| steps of 18 | 45 = 5 × 9 |
| steps of 22 | 55 = 5 × 11 |
| steps of 26 | 65 = 5 × 13 |
| steps of 30 | 75 = 5 × 15 |
| steps of 34 | 85 = 5 × 17 |
| steps of 38 | 95 = 5 × 19 |
Because numerator and denominator condition each other, the system is hard for a layperson to see through. But it continues without end, because it is nothing but the sieve, and it is at the same time the consequence of the tetractys, whose overlapping grids produce the seemingly chaotic distribution of the primes. No wonder two Greeks meet here, Pythagoras and Eratosthenes. The star figures of the simplexes show the same order when fanned out into the coordinate system (see Coordinate system).
Hebrew letters?
It has been remarked more than once that this matrix recalls Hebrew printed letters. The Hebrew square script, the “language of God”, replaced the old Hebrew script for religious texts in the first centuries AD, and medieval magicians used it together with star polygons such as hexagram and pentagram for their signs. Surviving amulets show star polygons on square script. Simplexes contain star polygons, and fanning them out into the division table gives exactly this matrix. This invites speculation, nothing more. All the more surprising that hardly anyone knows this pattern today. A short treatise exists at the Pythagoras Institute, a simplified version at divisorplot.com (see Links).
Final digits and Dirichlet’s theorem
Apart from 2 and 5, all primes end in 1, 3, 7 or 9. In the long run the primes are distributed evenly over these four final digits, over 40 types for two final digits, over 400 for three, and so on. This follows from Dirichlet’s theorem on arithmetic progressions. Gauss’s reported habit of hunting for primes in his head during boring surveying work leads to exactly such observations. These statements about final digits hold only in the decimal system, however. The relations of the tetractys invite us to think about Dirichlet’s theorem afresh, for Stoffel’s “four temperaments” too are basically such residue classes, only in a grid of 30 instead of 10.