Tetractys part 2 – the "disorder" of the primes
From 25 on, the multiples of 5, 7, 9, 11 … occupy the twin positions. Their grids start 10 apart and grow in steps of 4 – 10 : 4 = 2.5, like the pentagram.
Up to 24 the order of the twin primes is complete: every position 6n ± 1 holds a prime. With 25 this changes. 25 lies on a twin position but is divisible by 5 without remainder. Here begins the apparently irregular distribution that has given mathematicians so many headaches.
Two ways of thinking
The usual view looks for order within the primes themselves. After all, they are the building blocks of all other numbers. With computers and large sets of numbers one then finds an “irregular regularity”. Another route is the Riemann zeta function. According to the Riemann hypothesis, all its non-trivial zeros lie on a line with real part ½, and the distribution of the primes depends on these zeros. Might 2, the reciprocal of ½, be decisive for understanding the primes?
Grids ten apart
From 25 on, the twin positions are occupied by the odd multiples of 5, 7, 9, 11 … Each of these series has its own interval, and the series are strikingly ordered:
- The series of 5 starts at 25 and runs in steps of 10.
- The series of 7 starts at 35 and runs in steps of 14.
- The series of 9 starts at 45 and runs in steps of 18.
- The series of 11 starts at 55 and runs in steps of 22.
The starting values are 10 apart, the step sizes grow by 4 each time. And every time, start divided by step equals 2.5: 25 : 10, 35 : 14, 45 : 18, 55 : 22.
| 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 |
The grids of the odd multiples from 25 on. Red: a hit on a twin position, where a prime is missing. Green: numbers divisible by three, lying between the twins.
Every third grid hits numbers divisible by three instead of twin positions (the series of 9, of 15 …), and within each grid every third number is divisible by three. To understand the order you therefore have to include the odd numbers divisible by three outside the twin positions. The system then consists only of odd numbers, and what remains is the unbroken sequence of primes.
The distribution stays irregular. But there is a strict order that assigns the primes their places. The steps of four in one direction and the steps of ten in the other recall 1 + 2 + 3 + 4 = 10. The steps of four arise because the system consists only of odd numbers. The steps of ten are more interesting: on this reading, the distribution of the primes rests on a decimal code.
Five and 2.5
Take all multiples of 5 and divide them by the even numbers, and the result is always the same:
| 5 : 2 = 2.5 | 10 : 4 = 2.5 | 15 : 6 = 2.5 |
| 20 : 8 = 2.5 | 25 : 10 = 2.5 | 30 : 12 = 2.5 |
One has to ask whether five, together with two, plays a key role in fractions in general, independent of the decimal system. For 2.5 is also the core of two start figures: the pentagram is 5 : 2, every second point of a pentagon. The double pentagram is 10 : 4.
Pentagram (5 : 2) and double pentagram (10 : 4) have the same quotient 2.5, and 2.5 × 10 = 25, where the disorder begins.
Why five sieves out no primes
A seemingly heretical question arises. Why does the grid that starts at 5 and runs through all odd numbers not sieve out primes as well? If it did, there would be no primes apart from 1, 2 and 3. The answer lies in the first term of the sequence: the grid from 5 runs in steps of 2, 5 = 5 × 1. Five only hits itself.
And here a hint of the zeta function appears. Writing ½ as 0.5 gives:
| Fraction | × 10 | Start |
|---|---|---|
| 1 : 2 = 0.5 | 5 | 5 × 1 |
| 3 : 2 = 1.5 | 15 | 5 × 3 |
| 5 : 2 = 2.5 | 25 | 5 × 5 |
| 7 : 2 = 3.5 | 35 | 5 × 7 |
| 9 : 2 = 4.5 | 45 | 5 × 9 |
This decimal code is not an illusion of a mind that thinks in the decimal system. It is contained in the geometry of the simplexes, and geometry calculates in no counting system. The interplay of even and odd numbers produces, just as with the rhythm of six from triangle to hexagram, the doubling of the pentagram into the sacred ten of the Pythagoreans.
A literary trace
A find from 2010 is worth noting: a review of Thomas Pynchon’s novel Against the Day by the media theorist Friedrich Kittler. In the novel’s universe things announce themselves first in mathematics, and the review brings the tetractys and the Riemann zeta function together in thought. Whether Kittler was aware of a direct connection is not clear from the text.