Twin Primes

[axn]

Originally Posted by Hugh

First, how does one ever move from the situation in which there are an infinite number of gaps in the twin prime pair symmetry to a situation in which there are none if there is no such "last prime."

Be careful generalizing from the finite to the infinite. Your "symmetry" only works when you limit yourself to sieving with a finite # of primes (you get a repeating sieve pattern with periodicity of Pn#). The notion breaks down when we extend that to the whole (infinite) list of primes.

Originally Posted by Hugh

"All pairs of numbers will end up in a twin prime pair symmetry (either as the initial pair or by being cast out by another prime). Since, as we go up through the sieve, no prime can ever remove all the subsequent pairs, there must always remain more pairs ...

Which can be removed by higher and higher primes. It is not like there is any shortage of primes to sieve with. Going back to the point of generalizing from the finite to the infinite.

[science_man_88]

2. All multiples of these primes (eg composites that aren’t multiples of 2 or 3) can also be expressed as 6n ± 1.

e.g ? not i.e ?

never mind I messed it up to the point is usually eg. is used when there are multiple examples but ie means in english which means it works in all cases last i checked.

[Hugh]

Originally Posted by axn

Be careful generalizing from the finite to the infinite. Going back to the point of generalizing from the finite to the infinite.

Indeed, and I know that's always a danger.

You might be right. The thing I thought was interesting is not just the symmetry (which of course isn't a symmetry in primes or twins), but the fact that every iteration of the sieve leaves an infinite number of gaps.

Originally Posted by axn

Which can be removed by higher and higher primes. Your "symmetry" only works when you limit yourself to sieving with a finite # of primes (you get a repeating sieve pattern with periodicity of Pn#). The notion breaks down when we extend that to the whole (infinite) list of primes.

But going back to this for a moment, of course the symmetry must break down for the infinite list. Because eventually all "twin prime pair candidates" must become part of a symmetry. The point for me about a last prime is that since there is no prime that can remove all gaps in the pattern, there must always be gaps left that can only be filled by initial pairs being cast out, as well as by higher and higher pairs, no matter how high you go in the primes.

I'll ponder this point - either way thanks for the feedback.

[science_man_88]

Originally Posted by science_man_88

e.g ? not i.e ?

never mind I messed it up to the point is usually eg. is used when there are multiple examples but ie means in english which means it works in all cases last i checked.

forgot a point never mind I see what it means.

[Hugh]

Originally Posted by axn

Which can be removed by higher and higher primes. It is not like there is any shortage of primes to sieve with. Going back to the point of generalizing from the finite to the infinite.

Oh, also one more passing thought on this (feel free to shoot down).

It's not just the higher and higher primes we have to think about here.

The gaps in the first primorial-length symmetry can never be filled (not just in its infinite repetitions).

Now I know this primorial is getting longer and longer every time we add a prime to the sieve so you might argue it's another case of eliding the infinite and the finite. But again, it seems to me that you can never move on from this primorial containing a symmetrical pattern of twin prime candidates.

Eventually every pair will become part of a symmetry, and the only way the gaps in the first primorial symmetry can move closer to being closed is by higher pairs being cast out as initial pairs. This means that every time a pair is cast out, there will be at least one higher pair that can't be eliminated any other way.

In order for there to be a "last pair to be cast out" by being absorbed in a symmetry, they would have to be part of a symmetry in which all the other pairs would eventually be cast out by higher numbers. But this isn't possible. So there is no such last pair.

So you can't ever reach the "last pair to be cast out."

Any thoughts?

[Hugh]

Originally Posted by science_man_88

forgot a point never mind I see what it means.

Good.

ie and eg I've never been that sure about, not very mathematical language either way I guess...

[davar55]

Originally Posted by R.D. Silverman

It is total nonsense.

Please be specific on at least ONE point.

[axn]

Originally Posted by Hugh

The gaps in the first primorial-length symmetry can never be filled (not just in its infinite repetitions).

After sieving by Pn and looking at Pn#, we can then further sieve from Pn to SQRT(Pn#) to get the full list of twin primes in the Primorial range. I don't see how you can formulate the argument to exclude the primes from Pn to SQRT(Pn#) from consideration

Originally Posted by Hugh

Now I know this primorial is getting longer and longer every time we add a prime to the sieve so you might argue it's another case of eliding the infinite and the finite.

Indeed that is exactly what it is. I'll offer an analogy (doesn't really carry any mathematical weight, nonetheless useful when working with intuition). In any finite interval, the number of even integers is less that the number of all integers (roughly half). Yet, when we consider the infinite set of integers, it turns out that the number of even integers is exactly the same(*) as number of all integers!!! Intuition is not a particularly good tool to deal with infinity.

(*) Taking some liberties with language.

[axn]

Originally Posted by Hugh

In order for there to be a "last pair to be cast out" by being absorbed in a symmetry, they would have to be part of a symmetry in which all the other pairs would eventually be cast out by higher numbers. But this isn't possible. So there is no such last pair.

There is no "last pair to be cast out", since there are an infinite list of pairs. Remember -- the only assumption is that there is a last pair of twin prime (i.e something to be cast "in", not out).

In your infinity argument, is there any reason to suspect that any _particular_ pair after the assumed "last twin prime" can _not_ be cast out by a suitably high sieve prime?

[Hugh]

Originally Posted by axn

After sieving by Pn and looking at Pn#, we can then further sieve from Pn to SQRT(Pn#) to get the full list of twin primes in the Primorial range. I don't see how you can formulate the argument to exclude the primes from Pn to SQRT(Pn#) from consideration


Indeed that is exactly what it is. I'll offer an analogy (doesn't really carry any mathematical weight, nonetheless useful when working with intuition). In any finite interval, the number of even integers is less that the number of all integers (roughly half). Yet, when we consider the infinite set of integers, it turns out that the number of even integers is exactly the same(*) as number of all integers!!! Intuition is not a particularly good tool to deal with infinity.

(*) Taking some liberties with language.

Yes, infinity is always a slippery concept.

Though on the other hand, isn't there something a bit circular going on in your counterargument. I'm saying that no matter how many primes there are, there are still an infinite number of spaces in the TPP pattern. While you're saying "aha, but once we have an infinite number of primes there may not be an infinite number of twin primes".

The open question there is how the spaces in the pattern get closed. If they can only get closed by prime pairs being cast out, perhaps my argument isn't refuted.

Or can they all get closed multiples of primes? Well, I don't see how, since we know that the next prime never completes the pattern. The fraction in the proof never reaches zero, whereas 100% of prime pairs are in a symmetry.

So perhaps you're saying that your invocation of infinity trumps my invocation of infinity?

Really when we say "infinite" it could be argued that we mean "we never reach the end" - in which case "we never reach the end of the prime numbers" and "we never reach a point where there are no more twin primes" can both be correct.

I'm still not totally sure either way on this, though I'm glad you're focusing on this part of the argument. Always fun to think about infinity.

[Hugh]

Originally Posted by axn

There is no "last pair to be cast out", since there are an infinite list of pairs. Remember -- the only assumption is that there is a last pair of twin prime (i.e something to be cast "in", not out).

OK. There is an infinite list of pairs. They can be either "cast out" by a prime being added to the sieve when one of them is a multiple of that prime. Or they can be "cast in" when they become the first prime pair in a new symmetry.

The argument is that we can never "cast them all out". So there is always more that will have to be "cast in" by becoming the first pair of a new symmetry.

(Just as there is always another prime number that will become the first prime number in a prime "composite symmetry".)

Originally Posted by axn

In your infinity argument, is there any reason to suspect that any _particular_ pair after the assumed "last twin prime" can _not_ be cast out by a suitably high sieve prime?

Yes, the fact that no "suitably high sieve prime" can be the one to cast out the last remaining gaps in the pattern. That's the exact point I started from about the "last prime".