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Major Advance Towards a Proof of the Twin Prime Conjecture

Soulskill posted about a year ago | from the optimus-prime-conjecture-still-unresolved dept.

Math 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."

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First? (-1)

Anonymous Coward | about a year ago | (#43729029)

70 millionth post!

captcha: lessens

Re:First? (-1)

Anonymous Coward | about a year ago | (#43729197)

good to know that while the country is almost 17 trillion in debt that such important endeavors take priority

Re:First? (-1)

Anonymous Coward | about a year ago | (#43729327)

Good to know that while the country is almost 17 trillion in debt that you found the time to post on Slashdot.

Re:First? (0)

Anonymous Coward | about a year ago | (#43729515)

You sound mad.

Re:First? (-1)

Anonymous Coward | about a year ago | (#43729719)

You sound moronic.

Re:First? (1, Offtopic)

Chrisq (894406) | about a year ago | (#43729523)

good to know that while the country is almost 17 trillion in debt that such important endeavors take priority

What do you mean - they can now say "17 trillion dollars in debt is close to two dollars in debt when you consider infinity, the mathematicians say so!"

Preprints not avaiable, but it seems legitimate (5, Informative)

Anonymous Coward | about a year ago | (#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.

Re:Preprints not avaiable, but it seems legitimate (-1)

Anonymous Coward | about a year ago | (#43729101)

Who cares about nonsense such as this when I'm desperately searching for a rancid asshole! And you know what? I've found one; it's yours. Yes, you have quite the rancid rectum on you! My fetid, repugnant cock has taken a liking to your feces-infested asshole, and I'm going to jam it right in! What say you?

Re:Preprints not avaiable, but it seems legitimate (-1)

Anonymous Coward | about a year ago | (#43729129)

Someone got excited at the new XXX gay porn parody Anals of Mathematics.

Re:Preprints not avaiable, but it seems legitimate (0, Offtopic)

boundary (1226600) | about a year ago | (#43729143)

Truncate your search by looking in the mirror.

Primes are precious (1)

For a Free Internet (1594621) | about a year ago | (#43729041)

They should not be overused because they are a limited mathematical resource, like badgers or squid. I take care to use mostly non-prime numbers, especially even numbers because they are renewable. It is a little thing we can all do to protect our Nation's planet and its numbers for future generations of Americans and maybe for Filipinos too.

'2' - wrong, its 42 (-1, Offtopic)

nri (149893) | about a year ago | (#43729047)

everyone knows the answer is 42

Re:'2' - wrong, its 42 (0, Funny)

Anonymous Coward | about a year ago | (#43729055)

everyone knows the answer is 42

-1

Re:'2' - wrong, its 42 (0)

Anonymous Coward | about a year ago | (#43729081)

This. Slashdot definitely needs a "-1, Lame" mod.

Re:'2' - wrong, its 42 (2, Funny)

Anonymous Coward | about a year ago | (#43729163)

That would make it 41?

Re:'2' - wrong, its 42 (-1, Troll)

SupplyMission (1005737) | about a year ago | (#43729103)

what a stupid fucking response

Re:'2' - wrong, its 42 (0)

Anonymous Coward | about a year ago | (#43729553)

you obviously have never watched 'the hitch hiker guide to the galaxy'

Re:'2' - wrong, its 42 (0)

Anonymous Coward | about a year ago | (#43729565)

Watched? Hand in your geek card!

Not in North Carolina (5, Funny)

Anonymous Coward | about a year ago | (#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.

Stories like this... (2, Interesting)

Anonymous Coward | about a year ago | (#43729065)

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

Re:Stories like this... (5, Funny)

jamesh (87723) | about a year ago | (#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... (1)

Anonymous Coward | about a year ago | (#43729581)

Although funny, there seems to be some genuine wisdom bits.

Gaps between numbers... (5, Funny)

rew (6140) | about a year ago | (#43729099)

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... (0)

kyuubi (1355069) | about a year ago | (#43729195)

He's referring to the gaps between the primes, not the integers.

Re:Gaps between numbers... (2)

rew (6140) | about a year ago | (#43729215)

Yes. And I was joking, not serious. Duh!

Re:Gaps between numbers... (5, Funny)

Anonymous Coward | about a year ago | (#43729283)

Traditionally, when you're joking you should write something that's funny.

Re:Gaps between numbers... (5, Funny)

Anonymous Coward | about a year ago | (#43729393)

He may be English.

Re:Gaps between numbers... (1, Offtopic)

MadKeithV (102058) | about a year ago | (#43729419)

Traditionally, when you're joking you should write something that's funny.

Thank you mods for modding this comment funny.

Re:Gaps between numbers... (-1, Offtopic)

Barryke (772876) | about a year ago | (#43729459)

Traditionally, when you're joking you should write something that's funny.

Let me demonstrate.
  "You want me to fix your toaster? Hold it firmly and run hot water over it, while putting its plug in the power outlet."
This is only funny if you get it.

Humor is suggesting a (absurd yet understandable) relation between unlikely things.

Re:Gaps between numbers... (0)

auric_dude (610172) | about a year ago | (#43729639)

So am I, obligatory XKCD reference https://xkcd.com/899/ [xkcd.com]

Re:Gaps between numbers... (5, Informative)

Anonymous Coward | about a year ago | (#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).

Open set it is! (0, Redundant)

idunham (2852899) | about a year ago | (#43729107)

That means that prime numbers are an open set, since an infinite number of prime pairs with a finite difference means an infinite number of prime numbers.

In other words: If you wanted to be recognized for finding the highest prime number, you can stop your computer now. There is no highest prime.
But if you only care for a temporary entry, go ahead; you may well find one in a search of only 70 million numbers!

Re:Open set it is! (2)

Agent ME (1411269) | about a year ago | (#43729123)

It was already proved that there were an infinite number of primes.

Re:Open set it is! (5, Informative)

phantomfive (622387) | about a year ago | (#43729127)

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.

Re:Open set it is! (3, Informative)

Nyh (55741) | about a year ago | (#43729183)

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.

Re:Open set it is! (1, Insightful)

Anonymous Coward | about a year ago | (#43729227)

That was the point, yes. Are you unclear on the nature of the proof?

Re:Open set it is! (1)

wonkey_monkey (2592601) | about a year ago | (#43729229)

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.

Re:Open set it is! (4, Informative)

Nyh (55741) | about a year ago | (#43729275)

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

Re:Open set it is! (2)

z3alot (1999894) | about a year ago | (#43729313)

To be pedantic, we still cant conclude the product + 1 is prime, only that it is a contradiction that it is divisible by no prime (which is all we need anyway). The GP is correct in that a proof more similar to Euclid's original is given by considering an arbitrary finite set P of primes, letting N be the product of the P plus 1, and then concluding that a prime divisor of N must not be in P.

Re:Open set it is! (1)

bierik (575138) | about a year ago | (#43729385)

This isn't pedantic at all. The constructed number does not have to be a prime (as shown below in locofungus' example) and thus the additional step (there must be another prime, contradicting the assumption) is needed.

Re:Open set it is! (5, Informative)

locofungus (179280) | about a year ago | (#43729319)

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 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.

Re:Open set it is! (0)

Barryke (772876) | about a year ago | (#43729493)

Mod parent up. He got antiproof.

Re:Open set it is! (1)

evilbessie (873633) | about a year ago | (#43729625)

This whole proof is much easier if you use factorials as you can always prove there must be a prime bigger than N, as N! + 1 is not divisible by any number less than N. Which sort of gets around this weird way of attacking this problem y'all seem to be using which involves ephemeral 'set of known primes' which is weird in proofs.

Re:Open set it is! (3, Insightful)

locofungus (179280) | about a year ago | (#43729715)

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! (1)

evilbessie (873633) | about a year ago | (#43729789)

I was just objecting to the use of set of all primes, as you say there is no easy way to construct (or test) primes, however by showing that there must be a prime greater than an arbitrary value you have demonstrated there are an infinite number of primes* without requiring that you know all the primes in the first place.

*(N!+1)! +1 ad infinitum.

The proof in the article is that there exists an infinite subset of the primes where members are separated from at least one other member by less than 70 million. Which is a pretty hard thing to even get close to proving.

Re:Open set it is! (1, Insightful)

A Nun Must Cow Herd (963630) | about a year ago | (#43729651)

So you don't have the set of all primes after all... that's the point. The proof goes like this:
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! (0)

Anonymous Coward | about a year ago | (#43729543)

GP has not constructed a bigger prime. He has constructed a bigger number which may be prime or may be composite. The guy you're "correcting" is pointing this out, unfortunately in terms you're too mathematically naive to understand.

Re:Open set it is! (-1)

Anonymous Coward | about a year ago | (#43729309)

He said multiply all prime numbers. And since you're trying to prove that the number of primes is infinite this way you start by assuming they are finite in number. So you cannot have one that was left out.

Re:Open set it is! (1, Informative)

k.a.f. (168896) | about a year ago | (#43729399)

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.

No, the assumption was that you have all prime numbers. You're not allowed to violate assumptions within a formal proof, you're only allowed to point out self-contradictions.

Re:Open set it is! (1)

Anonymous Coward | about a year ago | (#43729521)

Can people stop trying to point out the flaw in this comment please? The poster is correct and if you look up the proof you will find this rider.

You, sir, are not a mathematician. The person you are replying to is because he gets why the stated proof was incomplete.

If the new number is not divisible by any of the primes in the initial set, then either it is a prime (contradiction - the initial set is not complete) or it is composite (contradiction - another prime exists outside the initial set).

The stated proof assumes the new number is prime. This is not a valid assumption.

Re:Open set it is! (2)

locofungus (179280) | about a year ago | (#43729531)

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.

Re:Open set it is! (1)

dkf (304284) | about a year ago | (#43729407)

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.

The operation itself is guaranteed to give you a number that is coprime [wikipedia.org] with the initial set. However, if you were to believe that there were a finite set of prime numbers and were then to use that finite set as the input into the coprime generator, you'd get something that is coprime with "all" prime numbers, which would therefore consequently show that there must be at least one prime number that is not in that set, and establish the result as a candidate for the missing prime. (If you previously believed that there are no prime numbers, the product-over-set-and-add-one operation produces 2.) The assumption that must have been false, given that everything else is a basic mathematical or logical operation, is that there is a finite set of prime numbers; there must be an infinite number of primes.

But TFA wasn't talking about this. It was talking about the number of pairs of primes where the difference between the pair is 2, and that's a very non-trivial property.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729589)

Nyh, I would personally like to thank you for correcting the original proof, and I would personally like to shoot the non-mathematicians who keep telling you you're wrong.

Sheesh, Slashdotters discussing math is like blind people discussing the Mona Lisa.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729189)

This isn't about whether there are an infinite number of primes. Rather it is a proof that there are an infinite set of pairs of primes which are at most a finite distance apart. This would mean that going into infinity the primes don't just get farther and farther and farther apart.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729611)

This would mean that going into infinity the primes don't just get farther and farther and farther apart.

That is implied but hardly proven.
This doesn't say anything about the distance between the pairs or the number of non-paired primes between those pairs.
The average distance between the primes could still increase as far as we know but if we look at prime density graphs we can guess that the average distance increment decreases.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729201)

That would give you an even number.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729245)

2 is prime, so you'd also multiply by 2. Anything times 2 + 1 is an odd number.

Re:Open set it is! (1)

Your.Master (1088569) | about a year ago | (#43729251)

No it wouldn't. 2 is a prime number. Any positive integer multiplied by 2 is an even number, thus the result of multiplying "all" primes is an even number. Any even number, plus 1, is not an event number. QED.

Re:Open set it is! (1)

c.r.o.c.o (123083) | about a year ago | (#43729255)

That would give you an even number.

Wrong. Multiplying all the known prime numbers will always give you an even number, so adding 1 to whatever result will give you an odd number.

Hint: 2 is a prime number. You figure out the rest.

Re:Open set it is! (1)

Anonymous Coward | about a year ago | (#43729247)

http://primes.utm.edu/notes/proofs/infinite/euclids.html

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729277)

Wait.
Not that I don't trust you or the proof, but how do you know that adding one doesn't result in an even number?
Or, in other words, how do you know that multiplying all the primes will get you an even number?

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729325)

Same AC here.
I got it now, since you end-up multiplying by 2, one of the primes.
Nice :)

Re:Open set it is! (3, Interesting)

mrvan (973822) | about a year ago | (#43729749)

Or more elegantly in haiku form:

Top prime's divisors'
product (plus one)'s factors are?
QED, bitches!

-- http://xkcd.com/622/ [xkcd.com]

Re:Open set it is! (1)

Nyh (55741) | about a year ago | (#43729173)

The Greek knew in 300 BC there are an infinite number of prime numbers. The same proof also shows the gab arbitrary between two numbers can be arbitrary large (even larger as 70 million).

This proof shows there are an infinite number of primes that are 70 million or less from each other.

Nyh

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729179)

That there are infinite prime numbers has been known since the old Greek.

If there was a finite number of prime numbers, you could multiply all of them. Call this number X. Now add one to X. Now add one to X. As the numbers that can be divided by a prime number P are always P integers apart (e.g. the numbers that can be divided by 3 are 3, 6, 9, 12, and so on. 6 - 3 = 3, 12 - 9 = 3 and so on). Therefore, of all the prime numbers that X can be divided by, the only one that X+1 can be divided by is 1.

Thus, X can only be divided by 1 and itself => X+1 is prime. But X+1 will always be larger than the largest prime, as X was found by multiplying every prime.

As X+1 cannot both be prime and larger than the largest prime number, there cannot be a finite number of primes.

Re:Open set it is! (1)

wonkey_monkey (2592601) | about a year ago | (#43729293)

Now add one to X. Now add one to X.

Oops. Just add one, not two ones. Two shalt thou not add.

Re:Open set it is! (3, Insightful)

locofungus (179280) | about a year ago | (#43729357)

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! (1, Informative)

Anonymous Coward | about a year ago | (#43729519)

His proof is fine, you're just not familiar enough with how proofs by contradiction work. It's true that 30031 is composite. Within the formal context of his proof by contradiction, it is simultaneously true that 30031 is prime. It is prime because it is not divisible by any other prime. It is composite because you found a non-trivial factorization. Both statements are true. That's why it's a proof by contradiction - there is a contradiction. Your wording is correct too, but it is not necessary to add the case that you did where X+1 is not prime, though that extra bit may be helpful for less mathematically experienced readers.

Re:Open set it is! (1)

Arrepiadd (688829) | about a year ago | (#43729817)

Actually, Euclid's proof [wikipedia.org] for the infinitude of primes says that the number itself is either a prime (which your example shows isn't always the case) or that the number can be factored by a prime not in the list provided (thus proving other primes exist). In your case both 59 and 509 are primes, showing the original list of primes was incomplete. Rinse and repeat.

Re:Open set it is! (1)

Arrepiadd (688829) | about a year ago | (#43729821)

Oh god, I saw the "--" in your comment and assumed it was the signature... only then I realized it was the final section of your comment. Sorry about that.

Re:Open set it is! (1)

sgunhouse (1050564) | about a year ago | (#43729305)

May. There is a trivial proof that there exist gaps larger than any given number ...

Pick any number n. Consider n! (that's "factorial", for the non-mathematicians). Now, n! - 1 might be prime (or not), but as n! is divisible by k for all k x and a prime q > p with q - p = 70 million, not that there will always be a prime within 70 million of x.

Re:Open set it is! (2)

sgunhouse (1050564) | about a year ago | (#43729355)

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, ...

Re:Open set it is! (1)

Splab (574204) | about a year ago | (#43729661)

Nope, still with the missing stuff there it doesn't make sense :-)

But that might just be because I'm slightly blind when it comes to mathematical proofs.

(Kinda like Zaphods sunglasses, but build into the brain, blacking out when stuff gets complicated)

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729413)

That means that prime numbers are an open set

Apart from everything else wrong with your statement that has already been pointed out, "open set"
doesn't mean what you think it means. It is a term from topology; a set is "open" if none of the points
on its boundary is a member of the set (like the interior of a circle, excluding the circle itself). Considered
as a subset of the real line, no nonempty set of integers is open.

Re:Open set it is! (0)

Anonymous Coward | about a year ago | (#43729505)

Not only are you forgetting the classical proof of the infinity of the primes (as pointed out passim elsewhere), you also don't know what an open set is.

An open set is a concept from topology, and means a set which does not contain its closure. It does not mean an "open-ended" set of (natural) integers. There's a word for those sets, and it is "infinite".

TFS (-1)

Anonymous Coward | about a year ago | (#43729147)

This is probably the worst written summary that I have ever read on Slashdot.

Re:TFS (5, Funny)

Raenex (947668) | about a year ago | (#43729295)

This is probably the worst written summary that I have ever read on Slashdot.

You must be new here.

Conclusion seems wrong (1)

Anonymous Coward | about a year ago | (#43729169)

That there are infinitely many pairs of prime number which are at most 70 million numbers apart does not guarantee that you'll find another prime number within 70 million from a given prime number. There are (infinitely many) prime numbers for which you'll find another one within 70 million, but it's not true for every prime number, so there can still be increasingly large gaps.

Re:Conclusion seems wrong (0)

wonkey_monkey (2592601) | about a year ago | (#43729361)

There are (infinitely many) prime numbers for which you'll find another one within 70 million, but it's not true for every prime number

Nope, it is true. Consider only the prime pairs. There are, we now know, no gaps larger than 70 million. Add the rest of the prime numbers and there can still be no gaps larger than 70 million.

Re:Conclusion seems wrong (0)

Anonymous Coward | about a year ago | (#43729483)

Maybe I wasn't clear.
This is wrong: For all P prime exists P' prime greater than P with P'-P<=70000000. (i.e. maximum distance between two consecutive primes is less or equal 70000000)
This is true: There exist infinitely many P prime such that there exists P' prime greater than P with P'-P<=70000000. (i.e. infinitely many prime pairs with distance less or equal 70000000)

Re:Conclusion seems wrong (0)

Anonymous Coward | about a year ago | (#43729537)

Nope, OP was right. The statement is that for any N, there exists primes p and q greater than N such that the difference between p and q is less than 70 million. The statement is not that if p is prime then there will be a prime in the interval between p and p + 70 million. In fact, by considering 70 million factorial, you'll see that the latter statement is false.

Re:Conclusion seems wrong (2)

locofungus (179280) | about a year ago | (#43729369)

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.

Re:Conclusion seems wrong (0)

Anonymous Coward | about a year ago | (#43729577)

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.

Have you considered the possibility that you might have misunderstood something? I mean, whenever I find that I have a trivial proof for something that has been an important open problem for decades (or centuries as in this case), the first question that I ask from myself is: "Why haven't the brilliant guys who have researched this problem before me found this solution?"

The usual answer is that my proof doesn't work. The second most common thing is that I'm proving the wrong thing.

Here, you are doing the second thing. You are proving that for each natural number n, there is a pair of primes with that many composite numbers between them. On the other hand, the article is proving that there's an infinite number of pairs of primes with a limited gap between them.

It's quite obvious that the pairs involved in your proof are different from the pairs involved in the article's proof.

Re:Conclusion seems wrong (1)

locofungus (179280) | about a year ago | (#43729781)

Have you considered the possibility that I might be replying to a particular comment in the parent my post was attached to?

The comment in particular that I was replying to had:

"but it's not true for every prime number, so there can still be increasingly large gaps."

To which my reply said:

"Not only can there be increasingly large gaps but there are increasingly large gaps."

Tim.

Re:Conclusion seems wrong (0)

Anonymous Coward | about a year ago | (#43729833)

You're the one that's misunderstood.

People seem to think this result means every prime is within 70 million of another prime. This is not the case. The grandparent tries to disabuse people of this notion by demonstrating that you can construct an arbitrarily long sequence of composite numbers, hence you can construct a sequence of 7,000,001 composite numbers, hence there must be a pair of primes that are more than 70 million apart.

The actual result, as you seem to realise, but many other people didn't, is that for any number there is some pair of prime numbers both greater and within 70 million of each other. There's room for both of these constructions because infinity.

Primes closer together? (1, Interesting)

allypally (2858133) | about a year ago | (#43729193)

Also means that there must be at least one prime in every sequence of 70 million integers.

Means I can put an upper bound on my prime search script....If it searches 70,000,001 consecutive integers and claims to have found no primes, I'll know the bugged little script is lying.

That's a helpful debugging heuristic. Thank you, Pure Math.

Re:Primes closer together? (1)

ledow (319597) | about a year ago | (#43729217)

I'm not sure it does.

"there are infinitely many pairs of primes that are less than 70 million units apart"

It just means that the individual primes in the pairs must be 70 million units apart (from each other) or less. (and where the hell did "unit" come from? Do they mean integer?)

Not that single primes must be. Not that one pair from the next must be. You could have twin prime pairs at any interval so long as the other half of the pair is within 70 million integers.

Unless I'm reading it wrong. The thing about maths is, you have to be VERY precise and not leap into assumptions without testing them very, very, very thoroughly first. 99.99999% certain isn't good enough for a mathematician.

Re:Primes closer together? (5, Informative)

As_I_Please (471684) | about a year ago | (#43729321)

Not quite.

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...

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.

Re:Primes closer together? (2)

Kjella (173770) | about a year ago | (#43729735)

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.

Re:Primes closer together? (1)

Anonymous Coward | about a year ago | (#43729387)

There is an easy proof that there can be arbitrarily large gaps between consecutive primes:

For example, suppose you want to find a prime gap of size at least 1000000. Let N = 1000000!, so,
in particular, N is divisible by every integer from 2 to 1000000. Then N + 2 isn't prime because it's
divisible by 2, ..., N + 157 isn't prime because it's divisible by 157, ..., N + 1000000 is divisible by
1000000. So there are 999999 consecutive composite numbers from N + 2 to N + 1000000. The
gap between the largest prime before them and the smallest prime after them is at least 1000000.

What the new proof shows is that as we go through the primes, the prime gaps aren't eventually
all huge: there will always be some further prime gaps that are smaller than 70000000.
This is perhaps more impressive when you realize that the Prime Number Theorem says that
the average size of the gaps does get as big as you like: for billion-digit numbers, the
average size of the gap between primes is about 2.3 billion. Nevertheless, close pairs of primes
will still occasionally occur.

Re:Primes closer together? (0)

Anonymous Coward | about a year ago | (#43729433)

Nope. Large intervals without such pairs are allowed, similar as that there is no prime pair with distance two in the range 90-96. However, there will always be a prime pair with distance two later on (e.g. 101,103) , if the twin prime conjecture is correct.

Similarly, if Yitang Zhang is right, even if you find a 1e20 big interval without primes (I did not check if this exists), after that, there will be a prime number followed by another prime within 70 million integers.

Re:Primes closer together? (0)

Anonymous Coward | about a year ago | (#43729645)

Following sibling post, the interval from N!+2 to N!+N is a range of N-1 numbers, all trivially composite. Set N = 1e20+1 and you are done.

Re:Primes closer together? (1)

bigg_nate (769185) | about a year ago | (#43729443)

Not quite.

What he proved is that given any number n, there exist prime numbers p and q that are (1) greater than n and (2) within 70 million of each other. For example, for a particular n, perhaps p = n + 1,000,000,000 and q = n + 1,069,000,000.

This is not quite the same as saying there's necessarily a prime between n and n + 70,000,000. In fact, since the average density of primes decreases as 1 / log(n), I'm pretty sure that statement is known to be false.

Amusing. (0)

idbeholda (2405958) | about a year ago | (#43729253)

I've been using a similar practice for a few years when implementing the API database for TT Livescan.

http://tot-ltd.org/API/ [tot-ltd.org]
http://tot-ltd.org/API/main.db [tot-ltd.org]

Use prime numbers 2 or higher to map API calls to a "number" specific family (add the collective values of the API calls from main.db, then convert the value to hexadecimal), based on the API functions (Windows 3.11 to Windows 7). The rate at which it can catch malware based on API calls alone is grotesquely efficient.

Conclusion wrong (4, Informative)

enriquevagu (1026480) | about a year ago | (#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:Conclusion wrong (0)

Anonymous Coward | about a year ago | (#43729629)

It depends. The exact interpretation of "the gaps between consecutive [prime] numbers don't keep growing forever" is true. There can not be an infinite sequence of increasing gaps between consecutive prime numbers. In any infinite sequence of gaps between consecutive prime number, there are infinitely many gaps which are less or equal 70000000. That means there is no point from which the gaps only increase.

However, the other interpretation, that the gaps grow for a while but are bounded, is wrong.

Re:Conclusion wrong (2)

tulcod (1056476) | about a year ago | (#43729721)

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

Re:Conclusion wrong (0)

Anonymous Coward | about a year ago | (#43729769)

To a computer scientist, easy doesn't mean you can write down an easy algorithm which does the job. To us it means that running the algorithm has to be feasible. There is no such algorithm for big prime numbers, hence the attention being paid to the search for ever bigger largest known prime numbers.

Cool Louis Vuitton Samsung Galaxy S4 Case (-1, Offtopic)

asoofai (2923785) | about a year ago | (#43729485)

The Louis Vuitton Samsung Galaxy S4 Case [ashinemart.com] is authorized, and the customer signs the screen with their finger to get the coupons to order the product with a discounted price. The receipt can be delivered by e-mail or text message, and goes out immediately. Each case is tagged with the location of the purpose. The whole process takes only a few seconds. This product is put under the Designer Samsung Galaxy S4 Cases [ashinemart.com] category, where you can find a lot of cases and covers for smart phones and digital devices.
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