In part 1 I discussed how it may be possible- although substantially difficult to find a formula that infinitely generates prime numbers using a computer.
But this has already been solved- partially.
I'm going to rephrase the question I asked before:
Is it possible to find a formula that finds infinite prime numbers using a computer script in a timely manner- where 'timely' is not over a million years?
Shown below is Willan's Formula:
$$ p_n = 1 + \sum_{m=1}^{2^n} \left\lfloor \left[ \frac{n}{\sum_{j=1}^m \left\lfloor \cos^2 \left( \pi \frac{(j-1)! + 1}{j} \right) \right\rfloor} \right]^{1/n} \right\rfloor $$
Using a computer with Willan's Formula, it would take an impossible amount of time to find every prime to hack into a bank. This is because Willan's Formula uses factorials:
!n
Factorials can be represented like so:
If n = 5 then,
!5 = 5 x 4 x 3 x 2 x 1
Which means that, for the 5th prime number, you would have to calculate 10! which is 3.6 million. Imagine the computation needed for the 20th prime.
Hint
$$ f(n) = n^2 + n + 41 $$
[ STATUS: DECRYPTED ]
[ ORIGIN: ALTERRA_CORE ]
Keep going!
When we are working with a massive prime, the numbers here become impossible to comprehend. The human mind, nor the computers we build, can comprehend a number more than the amount of atoms in the universe- which by itself is just an estimate.
The formula that I found last time- which is not mine- can calculate the first 40 or so primes easily. This isn't enough for Bitcoin breaking but it is a good start.
To go further, we need to visualize a couple things:
- A graph using a grid units and radian angles, where each square from the origin is +1 with a integer radian. Each square where both coordinates are prime (2,2) is a yellow dot.
- A graph where the aforementioned yellow dot is included but also a blue dot for double coordinates that are multiple of 6 eg. (6,6).
- Then, we have to show ourselves the gap between these numbers and also the multiples of 6 at least up to 1000.
Keep in mind this is a basic, barely functioning graph.
This below graph works better with more detail:
If we for now forget this working script, and stick to using the disappointing Desmos one, we can presume that primes organize themselves as a half of pi:
$$\frac{\pi}{2} \pm n$$
This is because following the visual patterns we have been provided by the graph forming a spiral shape, we can assume that;
$${\pi}$$
possibly forms here by being halved and plus or subtracted by a random variable represented by;
$$n$$
but factorials still have to be used.
$$\frac{\pi}{2} \pm n$$
The above is not a direct explanation, but it helps with understanding as it's close enough to be a profound discovery but still combines simple concepts that should have been learnt in high school.
$$y = \sin\left( \frac{\pi}{x} \cdot n \right)$$
How does this research relate to physics?
The patterns explored here are imitated in the laws of the universe, which are physics. Prime number spacing looks identical to the energy levels in the nucleus of a heavy atom (like Uranium).
If you wanted to go deeper into this, you could say that the randomness of prime numbers is linked to the simulation hypothesis and MUH - which is just a silly idea that popped into my head.
If the truth is that humans live in a reality bound by math and made by a being, then understanding that math would really be the most beautiful math.
By deciphering this formula successfully without the use of a factorial, we can build a working formula for infinite primes and also a Gap - 2 Function which I'll explain in the next part of this blog post.
For some reason, I find this number theory discussion extremely interesting despite not being able to comprehend most of the jargon that the internet spits out.
I don't even have a good background in math. To be honest, I have never really been interested in this topic anyways and I'm just honestly growing to be interested in science more seeing what it can achieve.
Thanks if you read to the end!