This is probably a wild stab and I'm pretty much just expecting as much criticism as I can get
I've had a go at the twin prime problem and here is what I came up with.
It might not be 100% accurate as I haven't been able to get much feedback on it .
Firstly I'd like to state that I'm assuming there is an infinite amount of prime numbers.
To make it easier to visualize and work with I'm going to use a wave function to describe the multiples of a number.
A sin function with a period of 2, starting from 2 will have its x intercepts at multiples of 2.
Starting from 2 as the first prime number, it can be seen that where the previous waves don't intersect the x axis
at integer values then there must be a prime number. For example 2 _ 4 are the intercepts of the sine wave of 2.
3 is a prime. At 5 there is no intercept so that is a prime.
Now the only reason the integers between the multiples of 2 aren't all primes is because they are multiples of the other
primes which I can say as the waves of other primes crossing those points.
It can clearly be seen that as the waves repeat (line up like they were before) then the conditions created by the waves repeat.
I.e., 6 is a common multiple of 2 and 3 and 7 is a prime. again 12 is a multiple of 2 and 3 and 13 is a prime, same for 19,
at 24 we see that the wave of the number 5 prevents 25from being a prime.
Now in the case above the recursion size is 6. The only reason this recursion doesn't create primes after every multiple
of 6 is due to the other waves also passing the numbers that follow multiples of 6 (simply put being factors).
Now for any amount of waves, there will always be a recursion size. Easiest way to find this would just be to cross multiply
the periods of the waves together, or you could find the lowest common multiple of all the waves, either way the waves repeat.
Therefore the initial conditions that caused the first twin numbers must also repeat. These conditions can be altered by the waves
of the newly generated prime numbers. The initial conditions for twin primes can be defined as that a number on either side of a
multiple of 2 is a prime.
Also it may be a bit unclear of what I mean by recursion size, recursion size refers to the conditions previous to the last formed
prime number (as in the waves present at the formation point).
Here lies two possible scenarios.
1. The new waves interfere in such a way that the initial conditions created by the recursion of the waves is impossible.
i.e. the conditions required to make twin primes are impossible.
2. The waves interfere but do not permanently get rid of the conditions required to make twin primes.
Lets assume the first scenario.
I'd like to state that the term step size refers to the minimum difference possible between two prime numbers, initially
we'll assume this to be 2 (as in the case of twins)
Assuming that there cannot be any more twin primes results in the step size increasing to 4. Again the density of the
waves intercepting will increase and the step size will keep on increasing. This doesn't mean that larger step sizes exist
just that there will be none less then the step size until it increases.
Eventually the step size will keep on increasing until it reaches the recursion size. If this does happen that means that
there is such a large density of waves that all the x integer intercepts in a stretch of the length of a recursion sized interval
are being passed though.
This would lead to no more prime numbers being formed as every single digit would have factors and the first premise
of there being infinite prime numbers must be false.
Therefore scenario 2 must be true.
Again I'm not sure at how well this is presented, there might be clear logical errors I did not see or maybe I wasn't
descriptive enough, but that was my crack at it.
If you got any of your own ideas on this problem post em ahead.