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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.
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.
It was already proved that there were an infinite number of primes.
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."
That would make it 41?
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
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.
I gather the comment system doesn't like all those symbols. It removed half of my reply. Let me try words ...
n! is divisible by k for all k less than or equal to n, so n! - k is divisible by k and (if k is not 1) is not prime. So n! - 1 to n! - (n + 1) are two numbers with a difference of n with no primes between them.
The result must show that for any x there are primes p and q with q > p > x and q - p less than 70 million, ...
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Not only can there be increasingly large gaps but there are increasingly large gaps.
(N+1)!+2 to (N+1)!+N+1 are N consecutive composite numbers - divisible by 2..N+1 respectively.
Therefore there are arbitrarily long sequences of composite numbers.
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
"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).
No! Why is this causing so much confusion.
I claim that SEO (Some enormous number) is the largest prime.
You construct 2*3*5*7*11*...*SEO+1 and claim that it is a prime not in my list.
I run a quick probabilistic primality test and prove that your number is composite. (which it almost certainly is)
Conjecture: There are no numbers of the form 2*3*...*P_{n-1}*P_n + 1 that are prime for P_n greater than 11.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Yes, it's more like this: Imagine if you took a sack of marbles and spread them infinitely thin, you'd expect that the distance between any two marbles to also grow to infinity. This is proof that primes are not like this, no matter how thin they're spread they'll cluster in pairs less than 70 million apart. The conjencture is that you'll always find another pair 2 units apart (like 5 and 7, 11 and 13 etc.) no matter how big the numbers get.
Live today, because you never know what tomorrow brings
Or more elegantly in haiku form:
Top prime's divisors'
product (plus one)'s factors are?
QED, bitches!
-- http://xkcd.com/622/
Yeah, that movie right?
Saying 42 might have been funny if they were researching a number that had some abstract relationship to the meaning of life - but even then it would be predictable and overused.
But it's not funny just to answer 42 to any mathematics question. It's not funny at all.
42
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?
Yes, it's more like this: Imagine if you took a sack of marbles and spread them infinitely thin, you'd expect that the distance between any two marbles to also grow to infinity. This is proof that primes are not like this, no matter how thin they're spread they'll cluster in pairs less than 70 million apart. The conjencture is that you'll always find another pair 2 units apart (like 5 and 7, 11 and 13 etc.) no matter how big the numbers get.
It would of course depend on _how many_ marbles there are and _how thin_ they are spread. In the case of prime numbers, there are still so many of them that we _expect_ two that are close together from time to time.
The number of primes If you pick a random integer around N, the chance that it is a prime number is about 1 / ln (N). If you pick an odd number, the chance is about 2 / ln (N). Now if you pick an odd number x, then the chance that x is prime is about 2 / ln (N), the chance that x + 2 is prime is about 2 / ln (N), both are not quite independent (if x is not divisible by 3, then x + 2 is more likely divisible by 3, same for 5, 7 etc. ), but the chance that (x, x+2) is a twin prime pair is roughly 1 / (ln (x))^2.
The sum of (1 / ln (x))^2 over all even integers x diverges; if you sum it for all x The same is true for every pattern like (x, x+2, x+6), or whatever pattern you choose, except for those patterns where it is obvious that these primes can't exist, like (x, x+2, x+10), where one of the three numbers must be divisible by 3.
Proving it is of course an entirely different matter. However, if there are infinitely many pairs of primes with a gap
So if you take the (x, x+2) conjecture aka twin prime conjecture, the (x, x+4) conjecture, the (x, x+6) conjecture and so on, which are _all_ assumed to be true, then we now know that at least one of them _is_ indeed true.
It can't be composite if it's a product of all of the primes plus one. It could only be a composite if it was a product of a subset of the primes, plus one.
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 ?
It occurs to me that my comment below is rather demonstrating your point, actually. If it's a composite then we've successfully proven that the complete set of primes is an incomplete subset of the complete set of primes, which is another contradiction.
No kidding!!! What do you say at this point?
Thanx xkcd!
On the one hand you take life too seriously, and on the other, you do not take playful existence seriously enough. Seth
p is any of the primes, x is the result of multiplying all the other primes in the list.
If px+1 was divisible by p, then px+1 = py, for some whole number y
dividing by p, y = x + 1/p. This cannot be a whole number as p >= 2 and x is a whole number.
I'm sure there's a better proof, but that's just off the top of my head.
Get free bitcoins: http://freebitco.in
He would be, but he doesn't get invited to those sorts of parties.
Out of modpoints but really liked a post? 1BDkF6TtmmeZ3yqXbz9yhdYVqRYnwFoXDj
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.
Euclid's Theorem in actuality does refer to the case where X+1 is not prime. It's essential to the proof.
It goes something like this:
---------
Take a finite list of prime numbers, A, B, C etc. (The assumption that they are "all the primes" is unnecessary.)
Find the smallest common multiple of them, X.
Add 1 to that.
The new number, X+1, is either prime or composite.
If it's prime, then that's it. We've generated a new prime not on the list.
If it's composite, then it is divisible by some prime, G.
Could G be one the primes (A, B, C. etc.) already on the list?
But remember, X is divisible by A, B, C etc. So if G is one of those primes, then that means that both X and X+1 are divisible by prime number G, which is impossible.
Therefore G would have to be a new prime, not on the list.
Now we have a larger list, A, B, C, G, etc. and can repeat the process.
We can always generate a new prime not on the list, and therefore the list of primes is without bound.
---------
There are two kinds of people: 1) those who start arrays with one and 1) those who start them with zero.
"Doesn't it make a lot of sense?"
No.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
Which is... back on topic... prime!
My present is the activity I am currently engaged in with the purpose of turning the future into a better past.
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.
That clarification was important. GP said:
> You now have a number that is divisible by none of the primes, which therefore must be a prime number
This is incorrect. The number must have a prime factor not in the initial list, which is a different (and more general) statement than "it must be a prime number."
The existence of a prime factor not in the original chosen set is proof that the set was not, in fact, all the primes. Thus you've shown that the original premise leads to a contradition, so the original premise is impossible.