Major Advance Towards a Proof of the Twin Prime Conjecture 248
ananyo writes "Researchers hoping to get '2' as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million. That goal is the proof to a conjecture concerning prime numbers. Primes abound among smaller numbers, but they become less and less frequent as one goes towards larger numbers. But exceptions exist: the 'twin primes,' which are pairs of prime numbers that differ in value by 2. The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart. He presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever."
Re:Open set it is! (Score:1, Insightful)
That was the point, yes. Are you unclear on the nature of the proof?
Re:Open set it is! (Score:4, Insightful)
Your proof as written is wrong.
I claim there are a finite number of primes viz:
2 3 5 7 11 13.
You construct 2*3*5*7*11*13+1 = 30031 and claim that this is a new prime in my list.
I say - no it's not 30031 is composite. (59*509)
--
The correct proof is to say that X+1 is either prime or is divisible by a prime not in the list thus proving that the list is incomplete. If the list contains all the primes up to N then there must be a prime bigger than N.
Re:Open set it is! (Score:2, Insightful)
1) suppose you have a set of all primes, and the set is finite.
2) show that there's another prime not in your set - that contradicts (1).
3) therefore, there is no finite set that contains all primes.
All you've done is demonstrate one example of step 2. The original proof given by phantomfive gives a different example of a prime not in the set. Either works - the proof is valid.
Re:Open set it is! (Score:4, Insightful)
I don't see why it gets around this problem.
The equivalent claim would be that
N!+1 is prime.
The correct claim is that N!+1 is prime or is divisible by a prime larger than N
The faulty proofs are trying to construct a prime not in the set. The correct proofs are showing that a prime exists that is not in the set without making any claims about what that prime is other than it's bigger than N.
I'm pretty sure that it has been proved that there cannot be a constructive proof that there are an infinite number of primes - i.e. there is no way to construct a prime larger than N for arbitrary N.
Tim.
Re:Open set it is! (Score:5, Insightful)
You've misunderstood the proof as a test to see whether a subset of primes up to prime n is complete. That's not the case. You start by taking the entire postulated finite set of primes.
The condradiction you receive - that it's possible to create a prime outside of the complete set of all primes - indicates that any finite set is incomplete. (Or alternatively that addition, multiplication, or sets work very differently than we assume, but let's stick to the form of mathematics the problem addresses.)
Re:Stories like this... (Score:5, Insightful)
Re:Stories like this... (Score:5, Insightful)
Re:Stories like this... (Score:5, Insightful)
Um, one question that a person could ask is: If this proof is found, how does it change the world? How would being able to use the proof influence something in the real world? I'm not saying it can't or won't, only that simply picking a brainy subject does not mean that doing things in it aren't basically intellectual masturbation.
The change to our world is this: we now know something that we didn't know before. Now we can teach this new knowledge to others (and by others I mean people smarter than me) who can find new places and ways to apply this new knowledge. They might never do anything interesting with it, or it might cause an avalanche of new findings, we don't know. But we, as a species, fundamentally know more today than we did yesterday.
As an example, the ancient greeks studied prime numbers. Was there any immediate use of primes at the time? Did it allow them to improve harvest? Defeat the Roman army? Nope, they just studied them. At the time there is no way that they could have conceived their application for encryption. Yet today, all commerce on the web uses the mathematics of primes.
It is not important to have an immediate use for knowledge.
Re:Stories like this... (Score:5, Insightful)
It's shoulders all the way down.
You were probably going for funny, but if I had mod points I'd call this insightful. It really is shoulders all the way down; no one accomplishes anything of significance without relying on many, many others.
Re:Gaps between numbers... (Score:4, Insightful)
The same result would hold for random numbers whose distribution gets more sparse with increasing N
This is false --- depending on how fast the random numbers "spread apart", you can have an infinite number of random numbers but a finite number of "close pairs". Simple example: for each positive integer N, choose N to be in your set of random numbers with probability 1/N. This gives you an infinite expected number of such random choices: sum 1/N over positive integers diverges. But what's the chance of adjacent pairs? The probability of N and N+1 being in your random set is 1/N * 1/(N+1). The expectation value for this set is *finite*: sum 1/(N*(N+1)) converges to a finite value.
Re:'2' - wrong, its 42 (Score:5, Insightful)
Some AC felt the need to make a lame '42' reference. Then, against all odds, it somehow managed to get back around to being on topic when someone else gave it a -1, thus rendering it a nicely prime 41. Then you came along and decided to be an ass. Well done.
But wait! With 41 you don't just get an "on topic" prime number. You'll also find that 41 is actually a twin in the twin prime pair of (41, 43)! That's right, it is completely on topic... so.... nah nah nahnah nah.
Now, as far as I can tell I've managed to make two relevant posts on the topic out of a seemingly impossible "42 duh duh" comment. On the other hand, you've managed only to be an asshole and contribute nothing other than bad karma. As far as you comment about making more money goes, I'm confused, who knows, maybe I got whooshed or missed a meme or something. Or maybe I've just been trolled. But, maybe you'd make more if you weren't such an asshole and instead just let people have a good time without trying to piss on 'em. Especially when it doesn't even matter.