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 . The interesting work then begins only at 25, where five disturbs the order of the twins for the first time.
The starting point: from 5 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 once all primes are switched on, 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. According to Aristotle the Pythagoreans already saw the universe as determined by 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. What is meant is the square script, in which every Torah scroll is written to this day.
Its Hebrew name Ktav Ashuri, “Assyrian script”, betrays its origin. It derives from the Aramaic imperial script that the Judeans came to know in the Babylonian exile and under Persian rule. From then on it gradually displaced the older Paleo-Hebrew script. The Dead Sea Scrolls are already almost all written in square script. Some put only the divine name YHWH in the old letters. On coins Paleo-Hebrew survived until the Bar Kokhba revolt (132–135 AD), and around 200 AD the Mishnah declared a Torah scroll in Paleo-Hebrew script invalid. The Talmud credits the change to the scribe Ezra.
It is called square script because of its form. Each letter fills an imagined square and is built almost entirely from horizontal and vertical strokes. Scribes hang the letters from a common upper line instead of standing them on a baseline. Only Lamed rises above the square, while Qof and the long final forms reach below it. That is exactly what recalls the matrix: there too the figures are composed of horizontal and vertical rows of equal cells.
3 mothers7 doubles12 simples5 final forms
The 22 letters of the square script in three rows of nine, from right to left, with their numerical values as units, tens and hundreds. The five final forms complete the third row. The second view shows the groups of the Sefer Yetzirah, the third the square in which each letter stands, with the upper line from which it hangs.
The letters are numbers as well. Aleph to Tet count 1 to 9, Yod to Tsadi 10 to 90, Qof to Tav 100 to 400. Since these are only 22 signs, a later count completed the hundreds with the five final forms up to 900, giving three rows of nine, a decimal system of letters. The smallest figure, the Yod, hardly more than a point, has the value 10. The Kabbalists say all other letters are formed from it, just as in the tetractys ten arises from the point.
The Sefer Yetzirah divides the 22 letters into three mothers, Aleph, Mem and Shin, for air, water and fire, seven doubles, assigned to the seven planets, and twelve simples for the twelve signs of the zodiac (see The Kabbalah). 3, 7 and 12 are the same numbers that recur before the throne of God and in the magic circle (see Christmas, the throne of God and the number 24).
Medieval magicians combined the square script with star polygons such as hexagram and pentagram, and surviving amulets show star polygons over Hebrew letters. Simplexes contain star polygons, and fanning them out into the division table gives exactly this matrix. This invites speculation, nothing more: that the script was derived from the matrix cannot be shown, and horizontal and vertical strokes are simply what is easiest to draw with reed pen and ruler. 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. That each of these last digits occurs infinitely often was shown by Dirichlet’s theorem on arithmetic progressions, the even distribution in the strict sense was proven by Charles de la Vallée Poussin in 1896. Gauss wrote in 1849 that since his youth he had often used a free quarter of an hour to count the primes in a stretch of a thousand numbers with the help of tables. From this he gained his conjecture on the density of the primes. 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.
Sources and further reading
Sources
- Nicomachus of Gerasa, Introduction to Arithmetic, around 100 AD, book I, ch. 13 (the sieve of Eratosthenes)
- Peter Gustav Lejeune Dirichlet, Beweis des Satzes über Primzahlen in arithmetischen Progressionen, Berlin 1837
- Sefer Yetzirah (Book of Creation), composed between the 3rd and 6th century AD
- Holger Ullmann, tetraktys.de, since 2010
Further reading
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford 1938, 6th ed. 2008