Yea basically it does involve Sieve theory I didn't know I was double counting until you pointed it out, thanks for that. I'll read more about sieve theory and Zhang and Maynard's work, its just that its a bit hard to read some symbols and figure out their meaning and how they fit in.
I don't think I can fix G(p) to be put as an equation to solve. What should I do with the post now? Should I remove the post, remove the bottom parts and improve the top, ask something different, leave it as it is, any suggestions?
@ErickWong I see that my question is not valid G(p) is must be less than $\frac{1}{3}$ I would be double counting a ton if it were greater. But if there exist a function G(p) that does removes all of the multiples it will need to be greater than or equal to $\frac{const}{\frac{(p+2)^2-1}{2}}$ as 'p' heads towards infinity. If G(p) is equal to that barrier there are a set of twin primes capped at 'const'. Thank you for the constructive feedback
@ErickWong now since you pointed that out I did have to re-think about G(p) but wouldn't you say the same thing would happen with the subtracting part? When I subtract (1/5) and (1/7) they meet up at (1/3*5*7). I am worried that since 1/3 is positive in the beginning not everything will cancel out
@ErickWong you just made a great point the range is $3$ through $(p+2)^2$ however G(p) always gives a lower bound that is how I think I defined it and if G(p)>1 there should be a lot more twin primes generated in the interval $p^2 and (p+2)^2$ rather than $3$ through $p^2$ like you said the function grows extremely slowwwly. I'll see if I did something wrong when defining G(p). And I'm referring to the twin prime conjecture.
@ErickWong as long as G(n) doesn't approach zero that's good news. Do you think if I can prove that this question is valid meaning that parts 1 and 2 are true can you prove that what your saying is true? If so, is there a chance a case can be made proving the conjecture? I wouldn't know how to publish a proof I could work with you if that sounds good, I don't have much to do until school starts in the next 10 days.
@JakeMirra The answer to the twin prime conjecture just appeared in the math as a question. I didn't mean to offend. I just wanted help, maybe a point the the right direction. I just feel like this is a different angle to tackle it in which might give an answer.
@JakeMirra The post is just my take at answering the twin prime conjecture. I'm no mathematician, I'm a college undergrad studying computer science. At first my goal wasn't to prove the twin prime conjecture I was just astonished with right triangles and their sides which led to patterns within the equations. I have proved them to myself with induction and direct proof. I wouldn't know how to start publishing one of the triangle proofs. From what I have seen in the programs I built with the math I did is that the math is right.
@GerryMyerson I deleted the older post right before posting this one. I know I should have just left it and edited there. I’m sorry I won’t do it again.
Its not just the sum of twin prime reciprocals it all primes excluding 2, plucking out 1 reciprocal at a time for each sum after. I can post another example because it does look like its just the twin primes.@WillJagy Thank you.
@GerryMyerson yes its the same idea I just reposted it with more detail. I guess it is just a rephrasing of is there an infinitely many twin primes answering it with partial sums of the reciprocals of the primes.
@JakeMirra My question is what does G(p) approach as p approaches infinity. I think adding more tables will help paint the picture and I'll explain the function even more through equations. I do feel like I rushed at the end I feel like the two halves need more connection. Thank you.
