It's really hard to figure out what question you are asking. I understand that sometimes a question takes a while to ask, because maybe you are pointing out some pattern, but you just launch into equations rather than taking a moment to establish motivation. You should put some thought into the structure of your post and try it again.

I upvoted your posting because of all the hard work that you put into it. That being said, I completely agree with Jake Mirra's comment. Re-expressing one of Jake Mirra's ideas, if you were looking at this query cold, would you be able to grasp the query's question immediately?

@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.

You already posted a question about that quantity $G(p)$, didn't you? You should link to it. Anyway, does this amount to anything but a rephrasing of the question of whether there are infinitely many twin primes?

@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.

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.

having two copies of the same post, even if one has more detail, is not a good thing for this website. It would have been better to edit the older post. AND PLEASE EDIT IN A LINK TO THE OLDER POST, AND EDIT A LINK TO THIS POST INTO THE OLDER ONE.

@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.

So this whole post is a veiled, "please fill in the details of my proof of the twin primes conjecture." Downvoted. You want to solve a centuries old math problem that has consumed entire careers of brilliant, hard-working mathematicians, get a PhD and learn the field.

@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.

@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.

@DerekPenaoza Your sum can be simplified significantly. The majority of terms come from pairwise products such as $ab+ac+ad+bc+bd+cd$. These can always be rewritten as $\tfrac12 [(a+b+c+d)^2 - (a^2 + b^2 + c^2 + d^2)]$, turning a double sum into single sums. Try to rewrite $G(p)$ this way and I think it will be more legible, plus we know a lot about sums like $\sum 1/p$ and $\sum 1/p^2$.

As written, I’d say your sum diverges to infinity as $p\to\infinity$, since it’s roughly as large as $\frac12 (\log \log p)^2$. However $\log \log n$ diverges extremely slowwwwwly, so it may be hard to observe this computationally (one can make a strong case that if you add reciprocals of every prime ever computed by human civilization, it would still be less than $5$).

Thank you @ErickWong I didn’t know I could do that to simplify. I’ll see what I could do to change it.

@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.

@DerekPenaoza What conjecture do you refer to? I am confident that this is still a very long way from proving twin primes conjecture, if that’s what you mean. For instance, if this is an estimate for the fraction of twin primes in a certain range, the fact that it eventually becomes $>1$ means that the approximation must get worse and worse as $p$ increases.

@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.

@DerekPenaoza This technique looks like a form of inclusion-exclusion counting. When you add back $1/(3\cdot 5)$ and $1/(3 \cdot 7)$ you may be double-counting $1/(3\cdot 5 \cdot 7)$, so superficially it has the appearance of being an upper bound, not a lower bound. Of course once you subtract out the third-order terms you probably get something that goes to $-\infinity$. But this is the heart of what makes sieve theory challenging :).

@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