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## Major Advance Towards a Proof of the Twin Prime Conjecture248

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."
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## Major Advance Towards a Proof of the Twin Prime Conjecture

• #### Preprints not avaiable, but it seems legitimate (Score:5, Informative)

by Anonymous Coward on Wednesday May 15, 2013 @02:30AM (#43729035)

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.

• #### Newbie question (Score:2)

I have a question: (excuse me for the realy bad formatting)

If the result of prime numbers (plotted), can be formulated as e^x, where Xaxis = numbers (zero to infinity) v Yaxis = [amount of unique distances observed] ;
and plotted against the plotting of prime numbers themselves ;
and plot a formula_3 that averages the coordinates that both euclidean functions output, towards infinity, through where they almost intersect ;
and form a formula_4 that equals the offset of the two euclidean function, relative to f

• #### Re: (Score:3)

"Doesn't it make a lot of sense?"

No.

• #### Not in North Carolina (Score:5, Funny)

by Anonymous Coward on Wednesday May 15, 2013 @02:40AM (#43729059)

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.

• #### Re:Not in North Carolina (Score:5, Funny)

on Wednesday May 15, 2013 @09:02AM (#43730671)
It's also against our state law that twin primes cannot cohabitate unless they're the same sex.
• #### Re: (Score:2)

So I guess in North Carolinian schools they only teach twin primes that consist of one odd prime and one even prime.

• #### Stories like this... (Score:2, Interesting)

by Anonymous Coward

Stories like this only remind me of how ignorant I still am and how I've wasted my life.

• #### Re:Stories like this... (Score:5, Funny)

on Wednesday May 15, 2013 @04:05AM (#43729397)

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!

• #### Re:Stories like this... (Score:5, Insightful)

on Wednesday May 15, 2013 @08:30AM (#43730425) Homepage Journal
So in addition to giants, there are great people standing on the shoulders of there own communities, too? That is somehow comforting.
• #### Re:Stories like this... (Score:5, Insightful)

<VortexCortex&project-retrograde,com> on Wednesday May 15, 2013 @09:50AM (#43731155)
It's shoulders all the way down.
• #### Re:Stories like this... (Score:5, Insightful)

<shawn-ds@willden.org> on Wednesday May 15, 2013 @10:51AM (#43731735) Homepage Journal

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: (Score:3)

Part of the art of humor is in being insightful while being funny. It why the greats are so great, they just tell you good life advice, but they frame it in an amusing way and we love them for it.

• #### Gaps between numbers... (Score:5, Funny)

<r.e.wolff@BitWizard.nl> on Wednesday May 15, 2013 @02:56AM (#43729099) Homepage

To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.

• #### Re:Gaps between numbers... (Score:5, Informative)

by Anonymous Coward on Wednesday May 15, 2013 @04:28AM (#43729491)

Joking aside, submitter is not a mathematician. This doesn't prove anything about the gap between arbitrary consecutive primes. That gap does indeed grow forever, by the known distribution of primes, but by "chance" one would expect a few pairs to lie close together. The proof is that this "chance" event still occurs as N tends to infinity. The same result would hold for random numbers whose distribution gets more sparse with increasing N so it just says that the primes are not "less random" than these (in a very informal sense).

• #### Re:Gaps between numbers... (Score:4, Insightful)

on Wednesday May 15, 2013 @11:03AM (#43731839)

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.

• #### Conclusion wrong (Score:4, Informative)

on Wednesday May 15, 2013 @04:06AM (#43729401)

"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).

• #### Re: (Score:3)

It /is/ very easy to calculate prime numbers. The apparently hard thing is to list them all, and to factorize non-prime numbers.

• #### Twin Primes (Score:3, Funny)

on Wednesday May 15, 2013 @07:13AM (#43730057)
My favorite example of twin primes is "867-5309/Jenny" performed by Tommy Tutone in 1982.

Thanx xkcd! [xkcd.com]

• #### Why do we believe the twin prime conjecture? (Score:2)

Do we have good reasons to think it's true? Or do we just see lots of twin primes and figure they never run out?

• #### Re: (Score:2)

Sort of, in that it would be really weird if it wasn't true. It is generally thought that primes follow a certain distribution (i.e. any given odd number has a certain probability of being prime, decreasing as the number gets larger). We know there are an infinite number of primes. That means this probability never reaches zero. If it is always non-zero, then there is also always some non-zero probability that we get two primes in a row (just the square of the probability). If there were not infinite t
• #### The Question (Score:5, Informative)

on Wednesday May 15, 2013 @08:59AM (#43730633) Homepage

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.

• #### not sure... (Score:2)

Ok, this isn't rigorous at all (obviously), but it seems to me that if the size of the gap continuously grew, but fluctuated randomly, you would still have an infinite number of primes close together even though the average distance between them never stopped increasing. They would become fewer and fewer, but never stop, and hence would be infinite.

Not doubting the guy's work, but I'm doubting the summary's "the gaps between consecutive numbers don't keep growing forever."

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