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Hello,

I posted previously but wasn't brilliantly prepared... I would like to thank those that fed back last time because it has actually really helped which hopefully you can see here - I would really like feedback and if anyone is able to help with the maths that would be amazing and I hope I've addressed the issues raised in the last post. I have created a dropbox folder of higher resolution graphics which I will paste at the end.

It all started when I saw this video: Primes and Twin Primes an awesome journey

Carlos was trying to find the millionth prime in an effort to win a prize and was using circles to do it… what he didn’t realise was that it was the millionth digit prime number.

He found that if he used circles then he could find primes to the square of the largest circle in the sieve - similar to the Eratosthenes from third century BC.

Here is an image of his work:

You can see that where:

ONLY a circle of diameter 1 & ONE OTHER number crosses the number line that number is a prime so a circle of diameter 3 is a prime…

He found that with a sieve of 142 (largest circle in the sieve is 142) you could “find” primes up to the square of that number 1422 = 20,164

Here’s what that looks like:

Here’s the sieve or the largest circle for just 142….

Here’s what that “data” looks like at larger numbers near the 20,000th

What I saw was that the gaps appeared to correlate with the electron configuration of atoms:

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10

Whose electron number sequence pattern is:

2, 2, 6, 2, 6, 2, 10, 6, 1, 10

and for a specific spin (1/2 the sequence above and the relevance I will explore later)

1     1     3     1     3     1     5     3     1     5 ...

Essentially electrons exist in “shells” called s, p and d etc and each electron is paired (with a few exceptions - cover that later) so electrons have either: UP SPIN & DOWN SPIN

Based on spin the electrons exist in groups of 1, 3 and 5:

This unusual pattern of 1, 3 and 5 appeared to match the unusual arrangement of electrons for spin up or spin down which is either a coincidence or something more.

If this IS connected I can logically assume that the arrangement would look like this where the spin down is on the left axis and spin up on the right - using Carlos circles for graphical simplicity:

I found that the prime gaps correlated up to a point at number 20 - so the first 40 electrons fit within the prime number sequence - pretty interesting.

But there is a problem with after 20 when the 4p6 and 3d10 are around the wrong way:

Number gaps in the prime sequence:

0     1     1     3     1     3     1     3     5     1     5

Electrons configuration spin up:

1     1     3     1     3     1     5     3     1     5

There seems to be a correlation between these sequences other than:

1     1     3     1     3     1     3     5     1     5  (Primes Gaps)

1s1 2s1 2p3 3s1 3p3 4s1 3d5 4p3 5s1 4d5  (Electron Configuration from Hydrogen to Cadmium)

So does that mean it ends there? no not quite… but we need to better understand how these electrons fill up the shells.

The current rule uses the Aufbau principle / diagram which itself falls down at 23… (https://www.chemedx.org/article/clarifying-electron-configurations)

When the 4s electron moves into the 3d…

and so another rule is used the Rydberg Rule which relies on the spectroscopic data and experimental data:

The Aufbau principle (theoretical model):

4s fills before 3d

1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<…..

The Rydberg rule (from experimental/spectroscopic data):

3d can fill before 4s / 4s electron moves to the 3d

1s<<2s<2p<<3s<3p<<3d<4s<4p<<4d<5s<5p<<4f<5d<6d…

Hence at 23 or Cr and Cu:

Cr (4s1 3d5) and Cu (4s1 3d10) where the 3d orbital is filled - or half-filled- prior to the 4s orbital:

At 23 the 4s shell empties into the 3d but why?

Here  are all the electron configurations and the red &b blue are where the Aufbau principle falls down:

Back to Carlos and the prime gaps - when I looked at his graphics I didn’t see tons of circles but more tons of sine waves…

So I changed his circles into  sine waves within a circle or sphere:

Instead of the circle of 2 we now have a wave of wavelength 2 and amplitude 1 (radius of the circle) and a circumference of 3.14

Adding the circles of 2, 3 or 5 we now have a wave of wavelength 2, 3 or 5 and amplitude 1, 1.5, 2.5 (radius of the circles)

And the same for  some prime factors turned into waves:

So what happens next?

Well if  we use the inverse square law that governs gravity, electrons, light etc as it moves from a source - in this case each time the wave passes the number line is considered a source...

and that wave follows the inverse square law that

So if 4 is the source then the area of the circle the energy is spread over is 12.57 at r = 1

As the wave at source 4 moves from 8 to point ? the area would need to increase to 50.27 (1/4 reduction in I)

So it would move to 16 and have a radius of 8…

(I've removed the smaller waves that could carry on to keep the diagrams clearer)

So here are some other prime factor waves growing:

6 (put the electron configuration back in)

8

9

etc

As the source increases in energy the wave increases…

This is all for spin up… what about spin down?

If you take your 4 wave and add in angular momentum and the wave is rotating at the origin:

If you take the wave 4 and rotate it on the origin then at 180 degrees the wave travels back the way it came and moves from 8 to -8 as the circle rotation flips from clockwise to anticlockwise:

So you now have a spin up, spin down for prime factors…

Here’s all the electron waves view:

You can see here that at the 4p and 3d there is “unusual” wave activity and that the 3d could very well fill first before the 4p even though the order is different.

You can also see that a single electro could exist in the 3d shell with a “loop back” at 4s and you can see why perhaps a 4s empties into a 3d.

Not all electron shells are on the same axis:

and if these are “electrons” there are some other elements that are pretty important.

Electrons are in pairs and these pairs are in groups and subgroups which are on a specific axis i.e.

So they don't all weave along the same axis…

BUT the a wave on the x axis would need to create waves that are also on the same axis… so they whole wave needs to be axis specific and these are the different axis.

The s - x axis

The 2p

On the x, y and z axis

The 3d

on the x, y, z and in-between the x, y and z axis

So the wave needs to link the right numbers on the right axis so it needs to be axis specific for the whole wave from source to the largest atom…

If you now add in the axis for each wave i.e. a wave that starts at 6 on one axis would need to cross the number line and be on the correct axis for later electron shells to this formation:

I have not done the f orbitals yet… BUT the 3d have an interesting arrangement with 2 of the 5 coming from previous axis and 3 new axis so you would expect to see 3 new waves at 3d - xy, yz and xz which you do…

So in an electro magnetic filed the magnetic field lines are perpendicular to the electric field:

This is hard to show on Keynote! So JUST the primes 2, 3 and 5

You can almost see the electric waves weave through the prime numbers which are acting as “insulators” like the plastic on a copper wire.

The primes are perpendicular to electric field waves 4 and 6… and the waves “weave” in an out with the primes between 2 field lines.

Apologies for the length... here is the link to the dropbox images: https://www.dropbox.com/sh/ur2da810bv2zjpv/AAAJdhMpa15wIEQnr7s74GuNa?dl=0

Any help with the maths or feedback would be great...

Thanks

Dan King

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