Prime Numbers

[NU_nutrition_TS]

This may be of interest to MJ23.

I've started this thread to avoid monopolising the Ancient Appreciation thread. Following on from what I posted on that thread regarding the challenge of finding the pattern in prime numbers (which you first raised, IIRC, and which inspired me to have a go myself), I think I am close to cracking this!

I first discovered a regular geometric distribution to prime numbers by creating a particular grid pattern that, by simple mathematical calculation, inserts all numbers in sequential order down columns and across rows when constructed on a spreadsheet. I then discovered, by further study of the pattern, that there was a consistently repeating, simple mathematical progression to the prime numbers - unfortunately you also got the odd false prime (25, 91, etc.).

I have now taken this mathematical progression and made another spreadsheet that calculates a series of primes for a given 'seed number'. The seed numbers are not primes but they also follow a consistent mathematical progression in line with the geometric distribution on the grid.

Tonight I have managed to create a formula that will filter out false primes, so I can now calculate only primes using the spreadsheet. The remaining problem is that you also generate duplicate primes. I hope to identify the underlying pattern for these and find a formula to eliminate them also. Eventually I should be able to generate an ongoing list of unique sequential prime numbers.

The final hurdle will be to streamline this into one workable formula - hopefully this can then be utilised on a computer without the need for a spreadsheet.

Unfortunately my advanced maths and computing skills are sadly lacking and I may not be up to this task! I hope I can crack it because there is a substantial cash prize for anyone who does - and a chance for their name to go down in the history books!

[MJ.23]

Seems like you've had a good crack at it Nu!

My maths is up to scratch but my computational skills are lacking. Ill stick to good ol pen and paper. Even if it takes me 10 times as long XD.

I have access to the one of the most advanced pieces of mathematical software on the planet. Maple 13. This is a serious serious piece of kit. Retails for £2000 + and is used by NASA to model space trajectories. Its a bit of a task to work though; requires a degree in itself practically! Im hoping this can help me along the way.

Or wolfram mathematica which i believe i can obtain from the university.

You might find this useful as well:

Wolfram|Alpha—Computational Knowledge Engine

Its a computational knowledge engine. Type in anything you want and it will work it out for you!! Very clever piece of programming this. It can deal with any formula you throw at it.

You may find the work of Sebastián Martín-Ruiz interesting. He converted the sieve of Eratosthenes into a prime number formula. However this formula does have constraints. This is a common theme amongst "prime number formulas" they tend to have very strict limitations associated with them.

Im having a brain storm over recurrence relations and Euclids algorithim currently. Havent been able to turn my mind to the problem much as ive had billions of assignments due in the last month or so.

Also you might want to brush up on your induction so if you do sift out a formula you can prove it for every integer n and n+1. This is where the cookie crumbles for nearly all prime number mathematicians!

[NU_nutrition_TS]

Thanks MJ for the very useful advice, insights and resource suggestions - I will be referring back to your post as I try to progress through this problem.

I doubt my computer skills are any more advanced than yours - and while I certainly feel a little more confident in my maths skills but I doubt I am anywhere near the mathematician you are!

My latest inspiration is that converting the 2D geometrical pattern I have discovered into a 3D model may further refine it and throw up more hints as to a definitive answer. This is something that can probably be accomplished on a computer but my virtual 3D modelling knowledge is basically nil, so I'm thinking about doing it in a real-world way with balls and sticks!

[MJ.23]

How are you getting on with this Nu? Come the end of the week ill really be able to have a crack at it. Caught up in the insane physics of photons at the moment. They are blowing my mind.

[NU_nutrition_TS]

Just when I think the answer is in my grasp it vanishes again! There is definitely a repeating pattern that can be visualised graphically but to put it into simple formulaic mathematical terms that can predict only primes from any positive integer is frustratingly elusive. I don't even know for sure whether the pattern I've uncovered is original to me or is already long known in the maths world - I find it hard to believe that no-one has ever seen it before, so I'm inclined to believe I have yet to find the original discovery on the 'net!

[MJ.23]

Ive had a hunch the last week or so linking prime numbers to radioactive decay. Now i think most people assume radioactive decay is completely random however we say its random as we cant prove otherwise. Using physical probability curves; in theory nothing in this universe is random.

I dont think anyone has tried linking these two fields before. Little is known about radioactive decay due to its supposed "random" nature.

The concept of random is another extremely difficult thing to grasp. It can help a lot with prime number theorems as our current lack of knowledge attaches a random label to prime number distribution.

My thought process is trying to place two "random" processes parallel to each other and seeing if there are relations. Should be interesting!

[MJ.23]

I have been brainstorming this once again over the last few days. I created a thought experiment based around Heisenbergs' uncertainty principle and Schrodingers' equation.

Imagine a particle (electron is the easiest) trapped in an n dimensional box with impenetrable walls.

Using a 1 dimensional box makes the maths a little easier to follow; but the concept extends into 2 and 3 dimensions.

On a large scale (say n= 1 metre) the classical laws of physics still apply and the electron will bounce back and forth off the walls without any issue.

However if we then shrink this box down to an order of n = 1E-9 -> 1E-15 m classical mechanics no longer holds and quantum theory takes over.

The electron itself will trace out a wave pattern in the box and we can use the psi wave function to denote this.

Using the Schrodinger equation:

This is for a particle trapped between a potential. Inside the box the potential will approximate to 0 (V=0) However outside of the box the potential will tend to infinity leading to a 0% chance of finding the particle outside the box. (So i thought at this stage; turns out the uncertainty principle is more of a ******* to work with than I remembered)

Schrodinger equation.jpg

Where h(bar) = h/(2*pi)

The solutions to this equation can be derived out as such:

psi(x) = Acos(kx) + Bsin(kx), Where k^2 = 2mE/h(bar)^2

This only covers the distance of the box itself (which I denoted as -l/2 and l/2) This keeps it symmetrical about the origin which is useful considering the cosine function is an evenly symmetrical function.

I can now assume the probability density function is also symmetrical, |psi(x)|^2

It may not be clear why I chose this infinite square well example; but this is why:

Using the wavelength of the wave that the particle traces out you can calculate the number of nodes that the wave has:

The reverse is also true as the mathematics is commutative. So I will place a prime number of nodes onto this electron. I took the lowest prime number, n = 2.

kL = n(pi) -> k = n(pi)/L -> K = 2(pi)/lambda -> lambda = 2L/n

Using this with n = 2:

Lambda = 2L/2 = L

In my example I used the length of L [((L/2)-(-L/2)) = L]

Therefore the wavelength is L

I then realised a big problem I did not take into consideration. (Linked to the uncertainty principle as mentioned above)

The probability density never truly reaches zero (very obvious mistake I should have picked up on. It is a bell shaped curve). I.e this means the particle can exist at any point either inside or outside of the box. They only have statistically preferred positions. The impenetrable walls are void due to the quantum tunneling effect.

In a picotorial form imagine a bell shaped curve with its tips extending past the box. These tips are governed by Heisenbergs' uncertainty principle:

(delta)X*(delta)P >= h(bar)/2

Therefore the electron can tunnel out of our universe, enter another one and jump back into ours at a different point in space if it so chooses. Incidentally this is my hypothesis for the faster than light neutrinos scare recently discovered.

This uncertainty destroys the credibility of the wavefunction as a means to predict any prime number.

What I was trying to do was contain n prime numbers within an infinite square well and be able to predict them using the wavelengths that the particles they represent would create. By classical mechanics this works up to a given point; but the foundational issue here is the violation of the conservation of classical energy which keeps cropping up in quantum mechanics. In QM it is plausible to have negative kinetic energy; which is strictly forbidden in classical physics.

Back to the bloody drawing board. I will try and stay out of quantum mechanics next time.

[NU_nutrition_TS]

Whoosh! That was the sound of all of that going way above my head!

I did have a bit of a rethink on prime numbers recently (but nowhere near the level of sophistication you show above). I guess I am still naive enough to believe any solution will be deceptively and elegantly simple - which is why it has remained elusive for so long!

[MJ.23]

You say that Nu; but you could well be correct. Some of the most advanced theories in physics reduce to incredibly simple formulas.

For instance, E=mc^2. The mathematics that leads to this formula is mind boggling. Einstein discovered several new fields of mathematics in his pursuit of this derivation. A good read is "Einsteins' miraculous year." It is a collection of 5 of his most important papers that changed the entire world. It gives a fascinating insight into his psychology.

Another example is Heisenbergs' uncertainty principle. I am currently reading a modern translation of the actual paper Werner Heisenberg wrote on the topic. It is 180 pages and all boils down to:

(Delta)X*(Delta)P >= h(bar)/2

The book is:

"The physical principles of the quantum theory."

Sometimes science may overthink elementary concepts when the answer is indeed infront of them.

If you are looking for a real challlenge then I would recommend:

"The principia mathematica" written by Isaac Newton. It is an astounding collection of his lifes work; and I have not yet fully completed it! I have had two attempts and get snowed under part way through. I doubt many people realise how brilliant Isaac Newton really was. His psychology is still a hot topic for modern books and literature. Imo the greatest scientist to ever live.