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Major Advance Towards a Proof of the Twin Prime Conjecture
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."
The paper seems to have been accepted by Annals of Mathematics, which is basically the number one mathematics journal.
Also, according to New Scientist, Henryk Iwaniec (a well-known analytic number theorist) has reviewed the paper and didn't find an error. This may or may not overlap with the review at Annals, though.
No siree. Ain't non prime numbers at all here in North Carolina since we done banned them. Ain't no angels felled out of the sky, ain't no computers breakin', and my cousin's kisses never tasted sweeter. Prime numbers are a godless socialist conspiracy against Jedus and mah wallet.
To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.
There is a simple, ancient, proof that there are infinite prime numbers.
Imagine that you did find all the prime numbers, every single one.
Then, take them, and multiply them all together.
Add 1.
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
"First they came for the slanderers and i said nothing."
GP has already used all the supposed finite number of prime numbers in constructing his contradictory bigger prime.
The proof constructs a number that is not divisible by any of the prime numbers in the set of all prime numbers. Therefore it proofs there are an infinite number of prime numbers. The conclusion the constructed number must be prime is wrong.
Nyh
This is probably the worst written summary that I have ever read on Slashdot.
You must be new here.
The GP's correction is right.
The GGP said that his number was prime. It might be, but it might not. But if it's composite then it cannot be divisible by any of the primes in his initial set so there must be a prime not in his set.
For example, if we assume 13 is the last prime then multiply them all together and add 1 we get 30031. But 30031 is not prime - it's divisible by 59 (which is a prime not in our set)
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Not quite.
This means that for every prime p such that p+q (where q is less than 70 million) is also prime, there exists another prime r bigger than p such that r+s (where s is also less than 70 million) is also prime. Note that there is no limit to the distance between p and r.
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
Don't feel bad. Maybe you've made coffee for, served fries to, or unclogged the toilet of one of these great people? Every little bit helps!
"the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever"
Actually, I disagree with the unfortunate writing of the sentence. The gaps between consecutive prime numbers are variable, and on average they DO tend to keep growing forever. This is a widely known result, the density of prime numbers decreases as the numbers grow. However, since the gap between consecutive primes is variable and it does not follow a regular function (otherwise, it would be very easy to calculate prime numbers), even with a very low density of prime numbers we can find a pair of consecutive prime numbers with a gap of only 2.
The problem under study is not wether the gap between consecutive primes keeps growing forever (which is true only on average, considering a long secuence of integers), but wether there are infinite such pairs of primes with gap 2. The new result found says that there exist infinite pairs of primes with gap 70M or less. However, this does not imply at all that no consecutive pairs of primes with gap > 70M exist (which, in fact, they do).
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.)
No kidding!!! What do you say at this point?
From a purely mathematical point of view you are incorrect.
The proof isn't that there's less than 70million units between each prime (like there's a lot of primes with a gap of two units eg 29 and 31, 41 and 43 etc). the proof is that there's in infinite number of prime pairs with a maximum of 70 million units between them.
You can still find gaps significantly larger. Those gaps are present between numbers that are NOT prime pairs.
eg: 29 30 31 32 33 34 35 36 37 39 40 41 42 43 44
Here there is a prime pair with a 2 unit gap between them (41 and 43), however the number 37 has a larger gap on either side, because it is not a part of a "prime pair". In your thinking you are excluding the primes that are NOT paried, and the gaps between where one pair ends and another begins. Each of which, according to the proof still has the ability to exceed 70 million units.
Disclaimer: I did not fully read the proof posted in annals of mathematics, but I'm pretty certain that this is the gist of it
--- To err is human... Am I more human than most ?
So in addition to giants, there are great people standing on the shoulders of there own communities, too? That is somehow comforting.
Researchers hoping to get '2' as the answer
In case anyone's as confused as I was, I think I've finally figured out The Question, which is:
What is the smallest gap between consecutive primes which occurs infinitely many times?
Or something like that. Everyone thinks it's probably 2.
systemd is Roko's Basilisk.
It's shoulders all the way down.
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.
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.
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.
Yes, but if you read the article, or hell, even the summary, then you'd know it was about primes.
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.
My present is the activity I am currently engaged in with the purpose of turning the future into a better past.