Tetractys part 1 – the order of the primes
All primes from 5 on lie in the rhythm of six, 6n ± 1. The numbers divisible by three form the axes of symmetry between them.
Mathematicians have often spoken of the primes with bafflement. Leonhard Euler thought them a mystery into which the human mind might never penetrate. Paul Erdős alluded to Einstein: God may not play dice with the universe, but something strange is going on with the primes. Julian Havil collects such remarks in his book Gamma (2003) and suspects a “determining system” behind the apparent randomness. The mathematician R. C. Vaughan summed up the dilemma: the primes are obviously distributed at random, only nobody unfortunately knows what random means.
In recent times, computer analyses of large sets of numbers have found certain regularities, for instance in 2016 the surprising preference of consecutive primes for particular final digits. A different approach is proposed here, one that also starts from the sieve of Eratosthenes: the primes are irregularly distributed, but the structure between them is strictly ordered. The system comprises all natural numbers, for how is one to recognise an order while ignoring a large part of it? It is an order that keeps confirming itself and that is confirmed by the geometry of the simplexes. Whoever engages with it notices that the series of numbers is nothing dull but something very much alive, a highly complex order that no human invented. It was also the source of inspiration for old thinkers such as Philolaus.
The number 1 and the definition
By today’s definition a prime is a natural number greater than 1 that is divisible only by 1 and itself. 1 is excluded so that factorisation into primes stays unique. Otherwise any number of ones could be added to every factorisation. Behind this lies the notion that the primes “generate” all other numbers.
This view deserves to be questioned, for already among the first three numbers there are peculiarities:
- 1 has the defining mark of a prime in its most extreme form: it cannot be broken down at all, its only divisor is itself.
- 2 is the only even prime. All the others, infinitely many, are odd.
- 3 plays a key role, which becomes visible below.
The rhythm of six
Write the numbers in rows of six, and all primes except 2 and 3 lie in just two columns: directly before and directly after a multiple of six. This is the rhythm of six, already described by Leibniz:
6n ± 1
It is why primes so often appear as twins (5 and 7, 11 and 13, 17 and 19 …).
The numbers 1 to 120 in rows of six. The primes (red) all lie in the first and fifth columns, on the twin positions 6n ± 1.
Wherever a prime appears on the number line, it is on one of these twin positions. From 25 = 5 × 5 on, however, more and more positions remain without a prime. These numbers, here called pseudoprimes, are divisible only by primes from 5 upwards, never by 2 or 3.
The remaining numbers are just as strictly ordered:
- Grid of 1: primes and pseudoprimes. They always stand as twins and are always odd.
- Grid of 2: even numbers not divisible by three. They too stand as twins.
- Grid of 3: all numbers divisible by three. They lie at equal intervals, form no twins and are alternately even and odd.
The numbers divisible by three thus form the axes of symmetry for both kinds of twin. The possible prime positions are arranged in mirror symmetry. This becomes vivid when the rhythm of six is wound up into a helix, six numbers per turn:
The rhythm of six as a number helix. All primes from 5 on stand on two vertical lines, the numbers divisible by 3 on two others, exactly in between.
The pitch of six numbers per turn is only for illustration, with any other pitch the facts stay the same. Such a number spiral has been published by Axel Klitzke, among others.
The first primes in the literal sense
In summary: 1 is the first number in the grid of twin positions, those divisible neither by two nor by three. 2 is the first number in the grid of even numbers not divisible by three. 3 is the first number in the grid of numbers divisible by three. So 1, 2 and 3 are the only primes that truly live up to the word prime, “the first”.
So there are two kinds of prime. All primes from 5 on lie on the twin positions of the rhythm of six, 2 and 3 do not, but 1 does. They also differ geometrically: all simplexes with a prime from 5 on as their number of vertices contain star polygons, the line (2) and the triangle (3) do not. For applied mathematics the usual definition of a prime has proved itself and is entirely fine. For research into the distribution of the primes, however, these two kinds should be distinguished, otherwise the definition blurs precisely the fact that matters.
The Pythagorean Philolaus already distinguished two forms of number, odd and even, and a third mixed from both: the “even-odd” (see The ancient sources).
Triangle and hexagram
Two of the four start figures are contained in this order: the triangle for the three groups of numbers, and the hexagram, the double triangle, for their mirror-symmetric arrangement in the rhythm of six. The two interlocking triangles show the interplay of even and odd numbers. The other two start figures, the pentagram and the double pentagram, belong to the disorder after 24.
Triangle and hexagram stand for the order, pentagram and double pentagram for the disorder of the primes.
Larger grids
The rhythm of six comes from 2 × 3. Before it there is only the grid of two, all odd numbers. Multiply by the next prime and you get ever larger grids: 2 × 3 × 5 = 30, × 7 = 210, × 11 = 2,310, × 13 = 30,030. In the grid of 30, too, all primes (except 2, 3 and 5) lie in fixed positions. But these grids quickly become unwieldy. Others have thought about the grid of 30 before, among them the artist and communication analyst Felix Stoffel with his “four prime temperaments”. What is special about the rhythm of six is that the complete architecture of twin formation is laid down in it. All larger grids show more and more gaps from 5 on. That is why the rhythm of six is the basis for part 2.
On this view the tetractys is the determining system of the natural numbers, confirming itself again and again because it is fractal. More under The key to the tetractys and The sieve of Eratosthenes.