binding the andat
December 1, 2013 4:19 PM   Subscribe

Closing in on the twin prime conjecture (Quanta) - "Just months after Zhang announced his result, Maynard has presented an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower." posted by kliuless (16 comments total) 38 users marked this as a favorite
That Wired article is a masterpiece of popular writing about mathematics. It's relatively comprehensible to folks with some college-level mathematics, but it's also deep enough to really convey the core of how the proofs work even to a layman. That's really damn hard, although it helps that the concept of twin primes is easy to explain to non-mathematicians (as opposed to, say, the Riemann zeta function).
posted by Nelson at 5:11 PM on December 1, 2013 [1 favorite]

Here's a related math problem that can be proved with high school arithmetic:

Prove that there is at least one prime gap for any arbitrary length n.
posted by empath at 5:26 PM on December 1, 2013

Seconded. Speaking as a non-maths person, that's a really great bit of explaining. Thanks for the post, kliuless!
posted by ardgedee at 5:33 PM on December 1, 2013

The YouTube channel Numberphile has a video about Zhang and prime gaps that is worth watching.
posted by I Havent Killed Anybody Since 1984 at 5:39 PM on December 1, 2013 [1 favorite]

This was a damn confusing theorum until I read the Wired article. Maybe. (Most authors don't manage to make it any clearer.)

IFF I untangled it correctly, it's (now) saying that for every prime you find out in the direction of infinity, there's another prime within 600 integers of it either way. FOREVER.

Talking about the twin-prime conjecture just confuses things because *this isn't about that*.

Correct me if I'm wrong, and I'll be more than grateful.
posted by Twang at 5:39 PM on December 1, 2013 [1 favorite]

If I'm reading it correctly, I think it's actually saying that there are an infinite number of primes, such that there is a second prime number within 600 numbers. It isn't saying that there is always a second prime within 600 numbers, just that infinitely many of them do satisfy the condition.

If they can get the number from 600 down to 3, then they've solved the twin prime conjecture.
posted by jenkinsEar at 5:42 PM on December 1, 2013 [2 favorites]

Hey empath, that doesn't seem like it can be right (unless I'm missing something which is probably the case).

Do you mean A - B = n for any integer, where a A and B are prime? That doesn't seem like it generates enough odd numbers, as they would need the to have the form A - 2 = n, which isn't true for all n.

Have I gone wrong somewhere?
posted by Ned G at 5:43 PM on December 1, 2013

A and b are prime, and N is the gap between, so yes, A-B is N and every number between is prime. For any number N, there is at least one gap that large. The proof will tell you how to find it, too.
posted by empath at 6:00 PM on December 1, 2013

That's right, jenkinsEar. To phrase it another way, the theorem says that if you keep looking at bigger and bigger prime numbers, you'll keep finding pairs that are within 600 of each other. Not every prime number is part of such a pair (in fact, they get rarer and rarer, according to the Prime Number Theorem), but you'll never run out of such pairs.
posted by ErWenn at 6:04 PM on December 1, 2013 [1 favorite]

But what if the last two numbers are both prime did you think of that.
posted by Joe in Australia at 6:13 PM on December 1, 2013

Let n be given.

Suppose there are only a finite number of primes less than n away from each other. That is, after some number X, all primes are at least n away from any other prime. So there is at most one prime in the range [X, x+n] and the average density of primes from then on is less than 1/n.

Therefore, n = 600.
posted by Phssthpok at 6:14 PM on December 1, 2013

i don't think so, empath. your first statement of the proposition falls apart when we consider n=7. since all primes are odd numbers except for the number 2, all pairs of primes would have to be an even number of numbers apart unless one of the numbers is 2. 2+7 = 9, which is a composite number.

your second statement of the proposition only makes sense if you meant to say that all the numbers in between a and b are composite. it is easy to prove that there are an infinite number of arbitrarily long sequences of consecutive composite numbers. x! - x is composite, x! - (x-1) is composite, x! - (x-2) is composite, and so on all the way down to x! - 2. this also works if you change the minus signs to plus signs and start with x! +2. i apologize if i misinterpreted your proposition.
posted by bruce at 6:18 PM on December 1, 2013 [1 favorite]

Not only is the article well-written, they went to the trouble to spell Cem Yıldırım's name with the correct Turkish "ı"s.
posted by benito.strauss at 6:23 PM on December 1, 2013 [2 favorites]

Just as a counter-example to the misreading of the proof, every number between 26293 and 26903 is composite (a gap of 610).
posted by 0xFCAF at 6:33 PM on December 1, 2013

Sorry, I did mean to say every number in between is not prime (which is why I said a prime gap to begin with) and your proof is the one I was looking for :)
posted by empath at 6:46 PM on December 1, 2013

OK, ErWenn, thanks. So this isn't saying that *any* prime we find has a 'neighbor' within N.

"... Pa ... Pn ... Pn+1 ... Pb ... ∞"
(Pn,Pn+1) distance ≤ N
(Pa,Pn) and (Pn+1,Pb) >N

Perhaps a stronger theorum will eventually sort out which primes DO/DO NOT have such a companion.
posted by Twang at 8:18 PM on December 1, 2013

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