The distribution of primes – a question of perspective
Why the tetractys should have anything to do with the primes, why professional mathematicians think little of the idea, and why it is still worth insisting on.
For professional mathematicians, linking the Pythagorean tetractys with the distribution of the primes is not up for discussion. Yet precisely this link is at the heart of the ideas presented here.
The obvious objection
One objection comes up again and again: common encryption methods rely on breaking large numbers into prime factors. Anyone who found a structure in the primes could break many keys. So many people would have an interest in this that it would long since have happened and been published.
The answer: the problem of the distribution of primes is to a large extent a problem of communication.
Many discoverers, little exchange
On this view, the geometric connections are well known to a small group of people but are not widely publicised. The number-theoretical connections are already contained in the sieve of Eratosthenes. So far the sieve has been used only as a tool for sorting out non-primes. Its significance has not been recognised. There are variants of the sieve that are useless for sieving but reveal much more about the character of numbers.
Several private scholars have each seen part of this order:
- Felix Stoffel published a classification of the primes in 2010, based on a grid of 30, into “four prime temperaments”. In doing so he found a variant of the sieve. Every correct system of prime gaps must contain the sieve, or it would be wrong.
- Peter Plichta developed the prime number cross and put the number 24 at its centre.
- A retired chief design engineer of a printing-press company also discovered a grid of 30 containing all prime positions.
What is overlooked: these grids are “decimally coded”. That only becomes visible once you leave the decimal system and take the geometry of the simplexes as a guide. If amateur researchers were more open to one another and worked together more, the puzzle would long since have been solved. Professional mathematicians, on the other hand, risk their reputations if they stray from the axioms of their discipline.
The architecture between the primes
The real substance of the objection lies elsewhere. Mathematics looks at the primes themselves. The numbers in between it notices only in passing. But to look at the primes alone is to blank out most of the picture.
The thesis: the distribution of the primes is governed by a system consisting of an infinite sequence of ever larger grids. These grids are formed by the numbers divisible by three and by the “empty” prime positions. Because all numbers are made of prime factors, the primes are regarded as the cause of all others. But an irregular distribution cannot be caused by the primes themselves. Symmetry is the supreme law of nature and physics, as a balance of forces. There are symmetries among the numbers too, and the fact that primes so often appear in pairs is one of them. If the primes are irregularly distributed, they cannot form the axes of symmetry themselves.
The image for this: the primes are not the building but what remains between its walls.
And a final thought: if a solution has been sought in vain for centuries, would a different approach not be in order?
Continue with The order of the primes.