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Das fraktale Muster der Primzahlen (captions in many languages)

A fractal algorithm shows prime number patterning.

Published in Mathematics

Dec 20, 2025

Birke Heeren

independent researcher, Birke Heeren

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This article focuses on number theory by modeling the deterministic, symbolic, and fractal patterns encoding prime numbers and their gaps, revealing structured prime generation beyond traditional statistical distribution theories. We present a novel mathematical machine with a novel fractal algorithm for encoding the pattern of prime numbers and gaps. This framework leverages three fractal processes “move”, “copy”, “change” with deterministic rules to construct six walksets. Walksets An and APn act as collectors, Bn and BPn determine prime or composite, Cn starts with all natural numbers and CPn contains the fractal process with always periodic patterns. Primes and gaps arise as emergent phenomena within this rule-based system – proving that this Ansatz holds for all prime numbers and gaps. We call this type of mathematical machine “Synchronous Factory Automaton” SFA. It was implemented in Java.

Video Transcript

Hello, I am Birke Heeren from Greifswald.

I invented a prime number sieve.

The Synchronous Factory Automaton , SFA for short.

The SFA contains an algorithm.

And now let's first take a look at what an algorithm is.

An algorithm is a step- by-step instruction manual.

There are different ways to represent algorithms.

Mathematical procedures,

how we will view them.

But also craft instructions with pictures,

Cooking recipes,

Descriptions in everyday language,

or code in a programming language such as Java.

How should an algorithm be designed?

An algorithm needs uniqueness,

Determinism, feasibility,

Termination and determinism.

There must be no contradictory statements.

and give unclear descriptions.

Determinism: For each subsequent step

There is only one possibility.

Feasibility: Every step must be feasible.

Termination: The algorithm needs to end.

Determinism: Given identical inputs,

The results must be the same.

We have 6 walksets for the algorithm.

We have the walkset APn , BPn and CPn.

Where P stands for " pattern ".

Then we have the walkset An, Bn and Cn.

And in Cn, there are all natural numbers at the beginning.

What exactly are walksets ?

Walksets are lists

which we can sort as we define it.

And the algorithm in Walkset CPn is fractal.

And we will now demonstrate what fractal is using an example.

We have a small Java program here .

That produces the Barnsley fern.

This is a mathematical, fractal fern.

And we have a window,

in which a fractal algorithm is called,

with different colors, different repetitions,

First slowly, then quickly

Before we start the fern, let's take another look at the algorithm.

These are x and y values,

which are in different equations

And " randomnumber " means

that via a random number,

This determines which equation is executed in each case.

And after each calculation, a point is marked.

Okay, now let's show the fern.

and press "Start".

The points are displayed slowly at first, and then quickly.

And now we see, we have a large fern leaf.

with many small fern leaves.

That means the small fern leaf is the same,

Like the large fern leaf, it is self-similar.

And on the small fern leaves

are even smaller fern leaves

and attached to those are even smaller fern leaves.

And this self-similarity, that is fractal.

The Cantor dust is also a fractal.

A fractal with only one dimension.

And I'll show you the Cantor dust,

because the SFA, the prime number sieve, that I invented,

also creates a one-dimensional fractal.

Now for the Cantor dust.

And that is, we have here – in the first step – a segment

and the middle third is wiped away.

This is the algorithm for the Cantor dust:

Wipe away the middle third in each step.

And you can see that he gets thinner step by step.

until only dust remains in the end.

Here we see the algorithm for the prime number sieve,

the Synchronous Factory Automaton SFA

It starts up here, with the " starting state ".

Here we have the 6 walk sets

and we always take a " step ", a step: n.

And then we'll do an update, so

such as the walksets An, Bn , Cn, APn and BPn

what it will look like after this step.

Then we find out if n is a prime number.

And then it is decided,

how CPn - the Walkset CPn - is renewed in the algorithm.

Once, if n is not a prime number.

And once when n is prime.

And we remember, an algorithm must always be terminated.

And this end is reached here at the arrowhead.

And that's where the algorithm stops.

And then you can decide whether to take the next step: i.e., another round or not.

Well, the fractal

develops in the Walkset CPn

and consists of symbols

What symbols do we have?

We can see that down here, on the blue background.

"L" stands for "live", everything is still quite uncertain,

whether it is a multiple, a prime number, or the 1st.

Number 1 is a special case here.

It is neither a multiple (it is not a multiple) nor a prime number.

That's why the number 1 has its own symbol, again the number 1.

The first step, n = 1.

We're going to do a little painting now.

We'll start here with the Astart walkset .

and ask ourselves: How does that become A1?

The rule of the algorithm is

The only element from B is added to the right side of A.

Since Bstart has no element, A1 remains empty.

Okay, let's try a different color.

How does Bstart become B1?

It takes the leftmost element, the one furthest to the left in Cstart.

and writes that in B1.

Thus, B1 has one element.

And at the same time, Cstart has already become C1,

It has lost its left element.

One could do that

to describe a production line in a factory.

And that's where the SFA gets its name (Synchronous Factory Automat)

The stock is here on the right.

This is where the coding takes place.

And there is the collection container for the finished product.

Step n = 2 to n = 5.

How does A1 become A2?

The 1 from B1 moves to A1 and

This turns it into A2.

How does B1 become B2?

The left element of C1 moves to B1, which is now empty, because the 1 has landed here.

And then we have another element: the 2nd.

And so C1 has also become C2.

because the number 2 is missing here.

Let's take the color yellow again.

How does A2 become A3? You probably already guessed it.

The 2 from B2 moves to A2 and is appended to the right. Thus, A2 becomes A3.

B2 is now empty, the 2 is now here

and gets the 3 from C2.

This is how B2 becomes B3.

And at the same time, C2 became C3, because the 3 is now missing.

In the fourth step.

If the 3 moves towards A3, it will join the right side.

and that makes it A4.

B3 is now empty; it takes the 4 from C3.

And as a result, B has another element, namely the 4th.

C3 has now become C4, because the 4 was cut off and it starts at 5.

And it's the same in the next step, the fifth step.

The 4 moves to the right end of A4, and so A4 becomes A5.

B4 is now empty, because the 4 is already here.

And B4 takes the 5 from C4, and so B4 becomes B5.

C4 became C5 because the 5 was cut out .

Those were the three walksets An, Bn and Cn.

Now let's take a look at the patterned walking sets .

pattern " is "muster ".

and that's why I sometimes call them pattern sets .

We have AP, BP and CP here.

You can see that the CP contains the pattern "L".

"L" stands for "live"

That means it is completely open whether the number

And because all natural numbers are included here.

and all natural numbers are still undetermined,

Here we don't say "L, L, L, ...", but "period L".

It is a periodic pattern consisting of a single symbol.

Now it is important that this production line of samples,

It runs completely synchronously with the production line of the numbers.

And these pattern symbols (L, M, P, 1) encode the numbers.

So that in the end, in the collection container, so to speak, you have

has the finished coding.

Which number is 1 here?

2 is P is prime .

1, 2, 3 is 1, P, P. Therefore, 2 and 3 are prime .

And so on. But how does that work?

We are now talking about the Walksets AP, BP and CP.

They contain symbols.

We had already looked at the symbols.

And the symbols encode the numbers.

It is important to have this production line of the factory

to keep it synchronized with the numbers production line.

That's what we're dealing with now.

I placed the game piece

here from the " starting" state " on the 1

In the Walkset BP, I've now gone to step 1.

CP " starting state begins with an "L".

And whenever I find an "L", the rule is,

that it is a prime number.

But number 1 is an exception!

It is neither a multiple nor a prime number.

Therefore, we encode them with "1".

And that's what happened here.

Now, the question is, what happens to the "L"?

How does that become "L" again here in CP1?

And for that we have three fractal processes: move , copy , and change .

And whenever there is an L,

All three processes take place:

(a period symbol should go above it here)

(because the L is always repeated )

"Move" means moving the first letter to the end.

to set the pattern, here it is also the only letter.

We have here: one. The pattern must exist once. That's all there is to it.

And change is also an exception with 1, because it does not make multiples.

from "L" again the pattern "L"

Now, let's place the game piece on the 2.

We find an L here. There it is.

And now that exception 1 is over, all rules always apply.

If we find an "L", we have a prime number!

So we encode the 2 with P as prime .

So we get from this L, here to the P.

And whenever we find an L

and have a prime number, the three find

fractal processes take place. So first, " move ":

We move the first letter to the end of the pattern.

Here we have the 2, the prime number 2,

Therefore, the pattern must be present twice.

We'll double it.

Make the period line longer, that's supposed to be one.

And then we count in " change ": 1, 2

Since we find an L, it has to go.

And that becomes a multiple of M.

That's how we get

here to the pattern

Now I place the game piece on the 3.

The rule is again: I have an L, so 3 is a prime number.

We see that here too: I have an L and

The number 3 is coded with P.

" move " is taking place now.

The first letter is placed at the end of the pattern.

We have the number 3 here, so the pattern must be present three times.

And the period line .

This should be a long line.

because this pattern also repeats itself.

And now, we still need to make " change ".

We count 1, 2, 3, that's already M. 1, 2, 3

and for that, an M there.

These are the multiples of 3

So we come to this,

to the pattern .

Well, we find here

an M before the 4

That means 4 is a multiple.

and then only the fractal process " move " takes place.

We had, I'll just show you that,

Here's the difference in what we do with CP when n is not prime and when n is prime.

And the difference is that if you find an "M", only " move " takes place.

But if you find an "L", it means " move , copy , change ".

So, now we'll do the " move "

and that's how we come

here in BP for the coding "M"

to the pattern .

And what's exciting now is...

BEFORE we move the game piece to the 5, i.e., take the next step,

can we already see that

The next prime numbers are 5 and 7, because

Here are the L's .

" L's " are (the candidates ), the candidates for prime numbers

and there are no more small numbers that could make them multiples.

So, one can already foresee something .

Now that we find an L here, 5 is a prime number.

Here too we find the L, the 5

is encoded as prim.

And we see that the pattern in CP, already

five times as long.

Too big for our playset here.

And that's why we're looking at,

Now, let me present one of my research findings.

where you can see the development in CP

a little further .

Here we see some files , outputs from my research.

The dots represent the elements in APn and BPn .

And the pattern, the L's and M's , is the pattern in CPn.

In CPn there are only L and M.

Codings (after 1 and after P)

They will only take place in BPn .

So that 1 and P only exist in BPn and APn , and these are only shown here as points.

Interestingly, whenever L is at the front, the pattern multiplies and

If M is in front, this does not happen in the next step.

Because then, of the three fractal processes move , copy , change, only move takes place.

Because multiple copies don't copy, and also not

to change something in the periodic pattern.

So, now we begin.

Now L is at the front.

M only shifts the pattern. The pattern no longer fits on the screen at all.

And that's it. We'll continue shortly.

Here we see an illustration.

The x-axis shows the prime numbers up to 100.

And on the y-axis, the size of the pattern is plotted in CPn.

And you can see with the prime numbers up to 100, the pattern in CP is already over 10 to the power of 30 symbols in size.

And there are so many symbols,

that it's difficult to calculate that on a computer anymore.

So, in most calculations, the man comes

only up to the prime number 17

with a few tricks even up to age 19 and

up to the prime number 29

That was the biggest thing I've ever achieved.

the purpose of the prime number sieve is to generate prime numbers,

That's not the most important thing at all.

It is important that the (SFA) sieve can do that,

But practically speaking, the point is not to generate prime numbers.

What the sieve (SFA) is good for and how it differs from the sieve of Erastosthenes

We will show how it differs or is similar in a moment.

Three of the walksets are graphically represented together as a sieve.

These are Bn , CPn and Cn.

Bn is wherever the number - the so-called step number -

It says where the green game piece was earlier.

And we see here from the starting point state , here 1, 2, step number 3.

Then CPn, where the LM patterns are located.

Here at the beginning, two Ls, then and so on.

And under the pattern, then Cn, the stock...

of natural numbers, which is still there.

In the beginning, all natural numbers are in the starting position. state ...

to infinity in Cn.

And then (in Cn) from the 2, from the 3, from the 4. Remember, the first number is always cut out.

At first glance, one might mistake it for a sieve of eratothenes ,

because that's how he wrote down the numbers.

But there are differences.

Firstly, the SFA always has a lower limit here,

but NO upper limit.

And that is a very important difference.

We have learned that an algorithm must also terminate; it must also come to an end .

and the steps must be feasible.

The Sieve of Eratothenes cannot be operated without an upper limit.

The sieve needs to be cut off somewhere.

because all multiples must be eliminated in the Sieve of Eratosthenes.

There are many videos about the Sieve of Eratosthenes on the internet; feel free to check them out.

And if the Sieve of Eratosthenes had no upper limit, one would never finish eliminating the multiples.

And then the step would not be feasible - not all the way through.

And it would not terminate; the algorithm would never end.

different here (in the SFA) . And why is it different? Because we have a PERIODIC pattern.

In which we can truly eliminate all multiples.

The periodic pattern (of the SFA), however large it becomes, is FINALLY FINAL!

A second difference is that in the Sieve of Eratosthenes...

... that one usually starts at one (by writing down the numbers) ...

...and then maybe up to 10, up to 20, up to 30 and so on...

... that you can arrange it in such a way that some of the things you delete appear as multiples one below the other , ...

... but then the order (in the Sieve of Eratosthenes) is disrupted ...

... and then you have to paint across it.

It's completely different here (in the SFA). Here there are only wonderful columns,

M-columns and L-columns,

which can be described by linear equations.

First, we'll focus on this L-column.

the trivial equation, f of x equals 1 x plus 2

You can see the number 2 at the top of the column.

Therefore, the plus 2 in the linear equation

and the sieve is one symbol wide and therefore

1 x (in the equation).

x is element N zero

and in linear equations (here in the SFA) N is always zero.

Let's move on to the next sieve.

The L-column with the equation f of x equals 2x plus 3

and the next one (equation)

f of x equals 6x plus 5

f of x equals 6x plus 7

You can see here too

This plus 7 comes from that,

The number 7 is in the header (of the column).

and the 6x comes from the fact that

The entire sieve is 6 symbols wide.

Prime numbers can only be found in the L columns.

The M columns contain multiples.

These two L-columns together contain all prime numbers greater than 3, up to infinity.

I'll now show you a diagram on the right.

Here (on this axis) we have natural numbers and here also (axis) x and (axis) y.

And with sieve width 2 we have an L-shaped column,

That means it's a linear equation. It's plotted logarithmically, which is why it looks curved.

With a sieve width of 6, we already have two L-columns and two linear equations.

With a sieve width of 30 we have 8 (equations) and even more (equations) with a sieve width of 210.

This means not only that the sieves (in the SFA) become much wider very quickly,

There are also more and more L-columns, and therefore more and more linear equations per sieve.

stayed until the end .

You can also read the whole thing , which contains many details that I haven't mentioned yet.

zenodo.org is a European server.

https://zenodo.org/records/17148139

A fractal algorithm shows prime number patterning.

You can read it here - online.

Or download it.

recommend " download ". NOT download all, because that's a lot.

I'm already on version 21.

Yes, that's it. Until next time.

Birke Heeren (She/Her)

independent researcher, Birke Heeren

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