If you take the first n primes. We'll call n=3. Then sieve the rest of the number line. You get rid of all the compsites of 2, 3, and 5. The very first composite that catches us by surprise isn't 2*7 or 3*7 or even 2*2*2*7, those are already accounted for. No the first unknown composite is 7*7. In this case p_(n+1) = 7. All the things that kinda look like primes below 49 really are prime.
So if I have some construction that sieves the first n primes the anything in that construction that is less than p_(n+1)^2 is a prime.
My problem is that I need to show that there is not one number but two that have a difference of two. There isn't anything in the definition of primes that insists they be there. I can count the number of twin primes in the whole set and I know the length of the set, so I can calculate the average gap between them directly.
That average says that there should be lots of twin pairs less than the square. But the twin primes are under no obligation to be evenly distributed in the first tiny sliver for my benefit.
Thus I need to say something else. In this case I am asking "how far apart can successive pairs get?"
Empirically it looks like the maximum gap is indeed less than the square of the next prime. So as far as I have results (n=22) I would be guaranteed at least 4 twin primes even if they were as far apart as they could possibly be.
The trend is actually increasing! Successive n seem to allow more and more twin primes less than the square.
But it is a very close thing. It grows slowly. Also I don't have a formula for it. The trend may reverse after Rayo's number or something.
I very much doubt it, we should be able to say something lame like "here is a super slow growing logarithm, that tracks the ratio of p_(n+1)^2/G(n). We don't know if the values are close to the curve but we do know that it crosses it infinitely often"
Daniel Wigton
npub1w4…q30zp
2026-07-21 23:27:05 UTC
in reply to nevent1q…4nzp
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npub1w4jkwspqn9svwnlrw0nfg0u2yx4cj6yfmp53ya4xp7r24k7gly4qaq30zpSeen on
wss://nos.lolPublished at
2026-07-21 23:27:05 UTCEvent JSON
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"content": "If you take the first n primes. We'll call n=3. Then sieve the rest of the number line. You get rid of all the compsites of 2, 3, and 5. The very first composite that catches us by surprise isn't 2*7 or 3*7 or even 2*2*2*7, those are already accounted for. No the first unknown composite is 7*7. In this case p_(n+1) = 7. All the things that kinda look like primes below 49 really are prime. \n\nSo if I have some construction that sieves the first n primes the anything in that construction that is less than p_(n+1)^2 is a prime. \n\nMy problem is that I need to show that there is not one number but two that have a difference of two. There isn't anything in the definition of primes that insists they be there. I can count the number of twin primes in the whole set and I know the length of the set, so I can calculate the average gap between them directly.\n\nThat average says that there should be lots of twin pairs less than the square. But the twin primes are under no obligation to be evenly distributed in the first tiny sliver for my benefit.\n\nThus I need to say something else. In this case I am asking \"how far apart can successive pairs get?\" \n\nEmpirically it looks like the maximum gap is indeed less than the square of the next prime. So as far as I have results (n=22) I would be guaranteed at least 4 twin primes even if they were as far apart as they could possibly be.\n\nThe trend is actually increasing! Successive n seem to allow more and more twin primes less than the square.\n\nBut it is a very close thing. It grows slowly. Also I don't have a formula for it. The trend may reverse after Rayo's number or something.\n\nI very much doubt it, we should be able to say something lame like \"here is a super slow growing logarithm, that tracks the ratio of p_(n+1)^2/G(n). We don't know if the values are close to the curve but we do know that it crosses it infinitely often\"",
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