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Mathematicians Team Up To Close the Prime Gap

Unknown Lamer posted about a year ago | from the there-can-be-only-4680 dept.

Math 194

Hugh Pickens DOT Com writes "On May 13, an obscure mathematician garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers. Yitang Zhang showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers, representing a major leap toward proving the centuries-old twin primes conjecture, which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13). Now Erica Klarreich reports at Quanta Magazine that other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower. By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million. Then Terence Tao, a winner of the Fields Medal, mathematics' highest honor, created a 'Polymath project,' an open, online collaboration to improve the bound that attracted dozens of participants. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower. Zhang's work and, to a lesser degree, Maynard's fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn't be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record. 'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."

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Mr President (5, Funny)

Anonymous Coward | about a year ago | (#45472025)

We cannot allow a prime gap!

Re:Mr President (1)

For a Free Internet (1594621) | about a year ago | (#45472111)

I say let's bury them under a barrage of boring even numbers! 8 8 8 8 8 8 8 8 8 8 8 10 8 8 8 8 810 8 8 8 8 86 256 8 8 8 8 12 8!

Re:Mr President (0)

Anonymous Coward | about a year ago | (#45472513)

We cannot allow a prime gap!

All prime suspects are to be extraordinarly renditioned and sent to Guantanamo.

regarding collaborative efforts (5, Insightful)

Anonymous Coward | about a year ago | (#45472037)

sometimes its better to go it alone, then come back to the group with your results so that someone else may profit from them.

sometimes its better to be a part a group in order to establish your ideas and discuss, then go it alone when the group holds you back.

Re:regarding collaborative efforts (5, Funny)

Russ1642 (1087959) | about a year ago | (#45472361)

Wow. That's like so deep man.

Re:regarding collaborative efforts (1)

Anonymous Coward | about a year ago | (#45472369)

Yeah, sometimes doing A is good, but other times, doing B is good.

Re:regarding collaborative efforts (2)

NatasRevol (731260) | about a year ago | (#45472489)

That's why C is always pissy.

Re:regarding collaborative efforts (0)

Anonymous Coward | about a year ago | (#45473165)

Both, A, B, and C are pissy, but not really.
That's D for you.

How to they prove that? (0)

Anonymous Coward | about a year ago | (#45472061)

Oh, nevermind...

Nice work (5, Funny)

Anonymous Coward | about a year ago | (#45472079)

If they keep this shit up, pretty soon they will prove that every number is prime.

Re:Nice work (4, Insightful)

pellik (193063) | about a year ago | (#45472099)

Re:Nice work (0, Troll)

Anonymous Coward | about a year ago | (#45472129)

And there in a nutshell is the whole anthropogenic global warming argument.

Re:Nice work (0)

Anonymous Coward | about a year ago | (#45472529)

Except for the time scale.

And the quantity of data samples.

Re:Nice work (1)

SleazyRidr (1563649) | about a year ago | (#45473241)

Wait, the infra-red spectrum of carbon dioxide is based on babies in anthropogenic nutshells? I'm confused.

Re:Nice work (5, Funny)

ebh (116526) | about a year ago | (#45472321)

3 is prime, 5 is prime, 7 is prime, 9 is bad data, 11 is prime, 13 is prime...

See Kuhn (3, Informative)

TheloniousToady (3343045) | about a year ago | (#45472085)

'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."

History is rife with examples of the lone genius making a leap forward, thereby allowing the crowd to take it even further. See The Structure of Scientific Revolutions [wikipedia.org] by Thomas Kuhn.

Re:See Kuhn (0)

Anonymous Coward | about a year ago | (#45472325)

I'm not saying his leap came from aliens, but... it was aliens

Re:See Kuhn (1)

Anonymous Coward | about a year ago | (#45472681)

Favourite quote re said crowd taking it further:

I am not a Kuhnian.
--Thomas Kuhn

Re:See Kuhn (5, Informative)

sfkaplan (1004665) | about a year ago | (#45473067)

Wait, what? If that's what you think Kuhn wrote, then you may need to go re-read the book.

His central claim was not that lone geniuses make leaps, but that leaps can rarely be attributed so clearly to a single individual, moment, or event. The Big Idea of that book is that the process of scientific advance is much messier, and much more contextually dependent, then we are lead to believe in popular accounts. Often the so-called "genius moments" are a critical step, but not easily or correctly identified as such until after the fact, making it hard to know *which* insight was really the critical one.

There's lots of dispute about Kuhn, but let's not make matters worse by incorrectly describing what he wrote.

Yawn. (2, Funny)

Anne_Nonymous (313852) | about a year ago | (#45472117)

>> which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13)

Yawn. Call me when you find a set of primes separated by one.

primes separated by one (5, Informative)

xxxJonBoyxxx (565205) | about a year ago | (#45472155)

Er...2 and 3. What do I win?

Re:primes separated by one (5, Funny)

MiniMike (234881) | about a year ago | (#45472281)

What do I win?

The tattered remnants of Anne_Nonymous's (probably not her real name) Geek Card, in a frame.

What is the greatest lower bound? (0)

Anonymous Coward | about a year ago | (#45472711)

The result is astonishing. Does anyone happen to know what the greatest known lower bound is? (i.e. the largest known difference of two successive primes?)
The prospect that one might eventually show the exact upper bound is even more astonishing.

Re:What is the greatest lower bound? (2, Informative)

Anonymous Coward | about a year ago | (#45472873)

There's none. the number of primes smaller than n is équivalent to n/ln(n) when n goes to infinity (thanks to Hadamard and Vallee Poussin theorem). If there was a upper bound for two successive primes, it wouldn't be the case.

Re:What is the greatest lower bound? (5, Informative)

ShanghaiBill (739463) | about a year ago | (#45472977)

Does anyone happen to know what the greatest known lower bound is? (i.e. the largest known difference of two successive primes?)

There is none.

Proof: Select an arbitrarily large number N. The numbers between (N! + 2) and (N! + N) are all composite ((N! +2) is divisible by 2, (N! + 3) is divisible by 3, ..., and (N! + N) is divisible by N). Since you can find an arbitrarily large span of composite numbers, there is no upper bound on the gaps between primes.

QED.

Re:What is the greatest lower bound? (-1, Redundant)

Dthief (1700318) | about a year ago | (#45473021)

MOD THIS UP!!

Re:What is the greatest lower bound? (2, Funny)

Anonymous Coward | about a year ago | (#45473259)

I call BS. That gap is only N-2.

Re:What is the greatest lower bound? (1)

SleazyRidr (1563649) | about a year ago | (#45473275)

It's people like you who make me want to learn more about maths. I felt like there could be an arbitrarily large gap between primes, but I had no idea how to express it. What you wrote, in addition that there are infinitely many primes proves the point perfectly.

Re:primes separated by one (1)

bill_mcgonigle (4333) | about a year ago | (#45472795)

A "misses the concept of infinity" patch to sew on your uniform.

But we all know the final lower bound will be 42 anyway.

Re:Yawn. (1)

beelsebob (529313) | about a year ago | (#45472159)

Okay... Found them. 2 and 3.

Re:Yawn. (1)

somersault (912633) | about a year ago | (#45472181)

{2, 3}

Re:Yawn. (-1)

Anonymous Coward | about a year ago | (#45472187)

Call me when you find a set of primes separated by one.

{2, 3}

:-)

Re:Yawn. (0)

ChristW (18232) | about a year ago | (#45472201)

2, 3

Thank you, I'll be here all week...

Re:Yawn. (-1)

Anonymous Coward | about a year ago | (#45472207)

2 and 3.

Yeah! Fields Medal coming my way!

Re:Yawn. (0)

magic maverick (2615475) | about a year ago | (#45472221)

I've found a set of primes separated by one.

{2,3}

Do I get a Fields Medal for that?

Also, mathematics is awesome. Even if I can't understand it.

Re:Yawn. (0)

Anonymous Coward | about a year ago | (#45472239)

2 and 3?

Re:Yawn. (4, Funny)

Anonymous Coward | about a year ago | (#45472263)

I can do better: I can prove that there are infinitely many pairs of prime numbers p and q separated by zero!

Here are the first few such pairs:

(2,2)
(3,3)
(5,5)
(7,7)

Factoring Primes (1, Interesting)

ebno-10db (1459097) | about a year ago | (#45472123)

Will they ever learn to factor prime numbers though? I understand it's difficult, but solving it would save a lot of embarrassment when people misstate the problem.

Re:Factoring Primes (1)

mrego (912393) | about a year ago | (#45472219)

Prime numbers are the scaffolding of the number system and all of nature. I am convinced that twin primes represent the spine or backbone of an understanding of prime numbers. If the mystery of twin primes fall, soon we will understand primes and then all of nature. And we'll be rich, evil, mad geniuses with unlimited power. Mhhhahahahaha

Re:Factoring Primes (1)

ebno-10db (1459097) | about a year ago | (#45472863)

we'll be rich, evil, mad geniuses with unlimited power. Mhhhahahahaha

Sounds good, as long as we don't actually achieve it. The joy of being an evil genius is in striving, not succeeding.

Re:Factoring Primes (0)

Anonymous Coward | about a year ago | (#45472267)

> Will they ever learn to factor prime numbers though?

It's easy: let p be a prime number. It's prime factors are: {p}

HTH

Re:Factoring Primes (2)

NatasRevol (731260) | about a year ago | (#45472541)

So much error. So much missing.

{p,1} are the prime factors.

Re:Factoring Primes (3, Informative)

Anonymous Coward | about a year ago | (#45472911)

According to modern mathematics definition, 1 isn't a prime number because it is invertible. If you allowed invertibles among prime numbers then uniqueness of the factorization in primes wouldn't hold anymore as your example shows. We could have {p} {p, 1} {p 1 1}.

Re:Factoring Primes (0)

Anonymous Coward | about a year ago | (#45472933)

Um, no:

https://en.wikipedia.org/wiki/Prime_factor

Re:Factoring Primes (1)

ebno-10db (1459097) | about a year ago | (#45472997)

Either way, leave it to a mathematical genius to ruin a joke.

Re:Factoring Primes (5, Informative)

Anonymous Coward | about a year ago | (#45472319)

Factoring prime numbers is dead easy. Here's an implementation in Python:

# factorize prime
# precondition: the argument is a prime
# if the precondition is not met, the result is wrong.
# result: The factorization of the argument.
def factorPrime(p):
    return [ p ];

It's the factoring of composite numbers that is difficult.

Actually, even factorizing composite numbers isn't really difficult. It's just difficult to do it in a way that finishes before you stop caring about the result. ;-)

Re:Factoring Primes (1)

ObsessiveMathsFreak (773371) | about a year ago | (#45473303)

It's the factoring of composite numbers that is difficult.

Actually it's even easier

def factornumber(n):
        return [ n,1 ];

(And now I can't want to see how someone out pedantics-me in continuing this petty-up-man-ship thread.)

blood sweat tears innocence (0)

Anonymous Coward | about a year ago | (#45472163)

all being confiscated, you can count #s on us but not much else

Even lower? (1)

srussia (884021) | about a year ago | (#45472165)

Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, (...) to push the bound even lower.

600 ought to be enough for anyone.

Re:Even lower? (1)

OakDragon (885217) | about a year ago | (#45472257)

A good joke, but I think you dropped some precision.

Problem solving abilities (5, Funny)

ebno-10db (1459097) | about a year ago | (#45472169)

Three people are asked to prove that all of the odd numbers are prime - a physicist, a mathematician and a programmer.

The physicist goes first. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a ... oops, experimental error, 11 is a prime ...".

Next the mathematician takes a crack at it: "3 is a prime, 5 is a prime, 7 is a prime, and the rest by induction".

Finally it's the programmer's turn. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime ...".

Re:Problem solving abilities (5, Funny)

VortexCortex (1117377) | about a year ago | (#45472337)

// [VC 2013.11.20] Fix primary oddity error in prime oddity test.
#define 9 015

Re:Problem solving abilities (5, Funny)

ebno-10db (1459097) | about a year ago | (#45472983)

An interesting paradox. You're not a real programmer if you realized that define was necessary, but you are a real programmer if you obfuscated it using that archaic octal notation.

Re:Problem solving abilities (0)

Anonymous Coward | about a year ago | (#45472339)

Next the mathematician takes a crack at it: "3 is a prime, 5 is a prime, 7 is a prime, and the rest by induction"

This part of the joke is playing on the two* different meanings of the word "induction".

This joke uses the kind of induction as used by empiricists: http://en.wikipedia.org/wiki/Inductive_reasoning [wikipedia.org]

In mathematics, a proof by induction has two parts. Part one is in your joke, the second part, the actual "induction step" where it is proven that what goes for n also holds for n+2, is missing. So this "mathematician" is not truly a mathematician: http://en.wikipedia.org/wiki/Mathematical_induction [wikipedia.org]

Yeah, I know, explaining a joke ruins it.

*Two meanings of "reasoning by induction" that is, leaving others meanings like "electromagnetic induction" or "induction of childbirth" out.

Re:Problem solving abilities (0)

Anonymous Coward | about a year ago | (#45472465)

Thanks, Sherlock.

Re:Problem solving abilities (2)

ebno-10db (1459097) | about a year ago | (#45472903)

I know, explaining a joke ruins it.

That point can't be overemphasized.

Re:Problem solving abilities (1)

SleazyRidr (1563649) | about a year ago | (#45473305)

Explaining a joke always makes it better.

Re:Problem solving abilities (1)

jalopezp (2622345) | about a year ago | (#45472503)

It's not a programmer, it's an engineer. A programmer what is proof.

Re:Problem solving abilities (1)

ebno-10db (1459097) | about a year ago | (#45472927)

It's not a programmer, it's an engineer.

You're right, I should have stuck to the original version. With the character as an engineer, it's a humorous error. With a programmer, it's more like what did you expect?

Re:Problem solving abilities (0)

Anonymous Coward | about a year ago | (#45472995)

Engineers don't prove things. They look them up in catalogs. They may just be stuck trying to find a prime number vendor that will sell them a 9.

Re:Problem solving abilities (1)

ebno-10db (1459097) | about a year ago | (#45473043)

They may just be stuck trying to find a prime number vendor that will sell them a 9.

No, that's quite easy. You don't believe all those specs, do you?

Re:Problem solving abilities (0)

Anonymous Coward | about a year ago | (#45472859)

Meanwhile, the politician says "If you vote for me, we will create a kinder, gentler society where *every* number can be prime!"

captcha: counts

Summary (1)

orlanz (882574) | about a year ago | (#45472203)

Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.

Re:Summary (1)

Anonymous Coward | about a year ago | (#45472351)

Zhang proved it's finite. The others have just lowering the finite number with newer proofs.

Re: Summary (1)

KramberryKoncerto (2552046) | about a year ago | (#45472357)

Didn't you see the phrase "lesser extent"?

Re:Summary (2)

wonkey_monkey (2592601) | about a year ago | (#45472381)

At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

It was Yitang Zhang who settled the original long-standing open question - that being, is there any number such that you will always find pairs of primes separated by that number or less. The ultimate goal is to solve the twin prime conjecture - bringing the number in question down to 2.

Your own wording is a little confusing - I'm not sure who the "eight" are, or whether "eight - James Maynard" refers to seven mathematicians, in which you couldn't describe them all as "an obscure mathematician" ;)

His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers

This (from the summary) is a bit of an ambiguous way to put it - it's not a bound on gaps per se, because there could still be consecutive primes separated by 70 million (such as 70,000,000! and it's neighbour), but there will always be another pair further along separated by less.

Re:Summary (1)

wonkey_monkey (2592601) | about a year ago | (#45472397)

(such as 70,000,000! and it's neighbour)

Err, ignore this. Getting confused.

Re:Summary (0)

Anonymous Coward | about a year ago | (#45472877)

It was Yitang Zhang who settled the original long-standing open question - that being, is there any number such that you will always find pairs of primes separated by that number or less. The ultimate goal is to solve the twin prime conjecture - bringing the number in question down to 2.

What's the problem? You can always find '3' and '5'. It's not like they hide or something.

Re:Summary (5, Insightful)

Kjella (173770) | about a year ago | (#45472657)

Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

No. Before May 2013 there was no proof on an infinite pair of primes being a finite bound apart.
- May 2013: Zhang, bound 70 million
- End of May 2013: Others, bound <60 million
- July 2013: Terence Tao & Polymath project: bound 4680
- Now: James Maynard, bound 600
- Twin conjencture: still unproven, bound 2

So the "big" discovery was Zhang, for managing to put a bound on it in the first place. The rest are improvements on that proof, but not very fundamental ones. Proving the twin conjencture would be huge, but nobody's done that yet. The Polymath project and probably many others are working on it. The conjencture is almost certainly true, but notoriously hard to prove. Probably the easiest "feel" to get for it is the Sieve of Eratosthenes, make a long list of odd numbers then strike out the multiples of primes. Once you strike out the 3s it'll be obvious you don't get triplets since 3, 9, 15, 21, 27 and so on are all multiples of 3 so the "candidates" are (5,7) (11,13), (17,19), (23,25) and so on. As you add more primes like 25 = 5*5 it'll get fewer and fewer pairs but they keep occuring rather randomly. It feels like that with infinite primes they'll randomly end up being next to each other an infinite number of times, but proving it is another matter. For example if you take the Fibonacci sequence (1,1,2,3,5,8,13,21...) it's obvious it's an infinite series but the distance between numbers also grows to infinity. Not so with primes, by these proofs.

Re:Summary (4, Insightful)

gnasher719 (869701) | about a year ago | (#45472665)

So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.

Here's what it real means: There were conjectures, one of them famous, which stated:

There are infinitely many pairs (p, p+2) of consecutive primes.
There are infinitely many pairs (p, p+4) of consecutive primes.
There are infinitely many pairs (p, p+6) of consecutive primes.
...
There are infinitely many pairs (p, p+600) of consecutive primes.

It is now proven that at least one of these conjectures is true.

momkind intends to keep us all on equal footing (0)

Anonymous Coward | about a year ago | (#45472255)

phony #'s be damnned

our spiritual centerpeace gets uncounted again? (0)

Anonymous Coward | about a year ago | (#45472315)

billions of innocents (all of us unchosens) in the unbalance

Twin primes ? (2)

ctrl-alt-canc (977108) | about a year ago | (#45472403)

They are not so sexy [wikipedia.org] , after all...

Re:Twin primes ? (1)

ebno-10db (1459097) | about a year ago | (#45472941)

Only a mathematician would think that two primes separated by six is "sexy". And they say programmers and engineers are a sad lot.

Need a summary of the summary (0)

Anonymous Coward | about a year ago | (#45472405)

Ok, I'm dense.. I don't get it. What are they saying exactly?
Are they saying we will always find a prime number within 600 of another prime number?
If that were true, you'd think we'd have figured it out with the help of computers by now?
I must be wrong in my understanding.

Re:Need a summary of the summary (0)

Anonymous Coward | about a year ago | (#45472545)

Ok, I'm dense.. I don't get it. What are they saying exactly?
Are they saying we will always find a prime number within 600 of another prime number?
If that were true, you'd think we'd have figured it out with the help of computers by now?
I must be wrong in my understanding.

Your understanding is correct. And no, computers can't help because well prime numbers are infinite.
At a certain your uber super quantum bio gel computer will run out of resources and you'll be none the wiser.

Re:Need a summary of the summary (2)

JTsyo (1338447) | about a year ago | (#45472551)

You will find 2 prime numbers within 600 of another prime pair.

Re:Need a summary of the summary (3, Informative)

Kjella (173770) | about a year ago | (#45473151)

No, the maximum distance grows without bounds. What this proves is that you can always find two more primes that are less than 600 apart, so the minimum distance does not grow without bounds. It has absolutely nothing to do with the distance between one pair of primes and another pair.

Re:Need a summary of the summary (1)

gnasher719 (869701) | about a year ago | (#45473251)

No, the maximum distance grows without bounds. What this proves is that you can always find two more primes that are less than 600 apart, so the minimum distance does not grow without bounds. It has absolutely nothing to do with the distance between one pair of primes and another pair.

A simple proof: If you take a large number n, then n! + 2 is divisible by 2, n! + 3 is divisible by 3, and so on until n! + n which is divisible by n. n! + 1 and n! + n + 1 might be primes, but none of the numbers in between. So we have a gap between prime numbers of at least n.

Re:Need a summary of the summary (1)

Zalbik (308903) | about a year ago | (#45473189)

No, that's not the theory at all....

The theory is that no matter how high you look, you can always find 2 prime numbers within 600 of each other.

i.e. For any number X, there exists a pair of prime numbers Y, Z where Z>X and Y>X and Z-Y600

It's entirely possible that having found Y,Z, there are no other primes anywhere near those two.

Re:Need a summary of the summary (5, Informative)

ledow (319597) | about a year ago | (#45472587)

That is, basically, the theory, yes. But if we can get that number down to "2" it proves a centuries-old conjecture that could lead to all sorts of interesting proofs of other things becoming true also.

In terms of computers:

You do realise that we use the difficulty of, in particular, finding large prime numbers as the basis for most modern security protocols implemented on computers? Precisely BECAUSE it's so hard to do?

We're not talking about 2, 3, 5, 7, etc. but we're talking about primes with MILLIONS of digits. Primes so large that even to prove they are prime can take a long time. Primes so enormous that multiplying two of them together makes a number that's almost impossible to factorise back to the correct original primes, so much so that we use it as the basis for things like SSL, TLS, etc.?

And, no, computers can't do mathematical proof. They can help as tools but they are dumb. You do not prove that every number to infinity has a prime within, say, 600 numbers by printing out every number. By definition you'll be there until infinity on even the fastest possible machine in the universe. You could prove that primes up to a number X that would hold true, but X would never be sufficient to prove it was always true. Just the fact that primes only have to be N numbers apart before you hit the next one could lead to mathematics that might well accelerate the discovery and manipulation of primes themselves.

But if you come up with a clever mathematical proof that GUARANTEES the answer, no matter what X is or how many billions of digits it has, without having to worry about ever finding a *particular* prime, then you have something that a mathematician can take as "fact" and incorporate into larger proofs about the universe. Imagine if we just assumed that every prime was like this, and applied it to a large scale engineering project, and then found out that actually - once you get past a few billion atoms - the premise doesn't hold? It'd be catastrophic.

The last "proof by computer" (i.e. by brute-force rather than as a tool) was the four-colour theorem. And even that was just because the problem could be reduced (by a mathematician, and using other proven theories, and logical inference) to a few thousand cases that the computer could iterate. It was used as a time-saver back in the days of manual calculation, not mathematical proof.

Re:Need a summary of the summary (4, Informative)

smallfries (601545) | about a year ago | (#45473095)

No we don't.

Primality testing is easy - the problem is in P. Approximate methods for finding primes are very efficient. Exact checking is rarely used.

Modern security protocols rely on the problem of factoring a number into primes being difficult. Or on inverting exponentiation within a prime field.

Re:Need a summary of the summary (5, Informative)

x_IamSpartacus_x (1232932) | about a year ago | (#45473125)

It looks like you don't understand what GP was asking (at best) or you don't understand the summary/primes.
I think the GP was asking if there are always less than 600 between primes. The answer to his question is "no". The higher you go the larger gaps can be between primes. There can be untold millions/billions/trillions etc. between two distinct primes. This proof shows not that there are never more than 600 between primes, but that there are an infinite number of pairs of primes that are separated by less than 600. The difference is small but important. There may be two primes separated by a vast number, yet the higher you go there will always be a pair of primes coming up that are separated by less than 600.

For example:

The numbers
2^57,885,161 - 1
and
2^43,112,609 - 1
are primes. They have 17,425,170 and 12,978,189 digits in them. They are the largest two primes we know of. They are separated by a bunch of numbers in between them, almost 5,000,000 DIGITS (note digits not numbers) and all the numbers between them are composites. HOWEVER, the next largest prime may simply be (2^57,885,161 1) + 600 because there will always be a chance that there is a prime coming up less than 600 away from the current highest prime.

This is getting closer and closer to proving the long held belief/hope that there are an infinite number of primes separated by only 2. NOT that EVERY prime is separated by 2 from every other prime. That would be obviously false. Simply that there are an infinite number of primes salted throughout all those impossibly high ones that are only 2 apart.

Re:Need a summary of the summary (5, Informative)

Warbothong (905464) | about a year ago | (#45473171)

And, no, computers can't do mathematical proof. They can help as tools but they are dumb. You do not prove that every number to infinity has a prime within, say, 600 numbers by printing out every number. By definition you'll be there until infinity on even the fastest possible machine in the universe. You could prove that primes up to a number X that would hold true, but X would never be sufficient to prove it was always true. Just the fact that primes only have to be N numbers apart before you hit the next one could lead to mathematics that might well accelerate the discovery and manipulation of primes themselves.

But if you come up with a clever mathematical proof that GUARANTEES the answer, no matter what X is or how many billions of digits it has, without having to worry about ever finding a *particular* prime, then you have something that a mathematician can take as "fact" and incorporate into larger proofs about the universe. Imagine if we just assumed that every prime was like this, and applied it to a large scale engineering project, and then found out that actually - once you get past a few billion atoms - the premise doesn't hold? It'd be catastrophic.

What you say has nothing to do with computers. Why would anyone program a computer to work case-by-case like that? It's just as futile as going case-by-case by hand. Likewise, if one is inclined to generate higher-level, logical proofs by hand, then why not program a computer to generate higher-level, logical proofs? Oh wait, that's been done for decades (eg. AUTOMATH, or the entire field of Automated Theorem Proving).

The last "proof by computer" (i.e. by brute-force rather than as a tool) was the four-colour theorem. And even that was just because the problem could be reduced (by a mathematician, and using other proven theories, and logical inference) to a few thousand cases that the computer could iterate. It was used as a time-saver back in the days of manual calculation, not mathematical proof.

Erm, what lets you define "proof by computer" as "by brute-force"? Are you claiming that all computer programs are brute-force? That's clearly nonsense. Are you claiming that a computer running an efficient algorithm is just a 'tool' and that the Mathematical ability actually exists in the algorithm's programmer? If so, you must also claim that Deep Blue's programmers are much better chess players than Deep Blue. In that case, why weren't they the world champions?

Also, the brute-force 'proof' of the Four Color Theorem, from 1976, was unsatisfactory to many people. It only proved the Four Color Theorem under the assumption that the program is correct, but nobody could verify such an assumption. In 2005 a new proof-by-program was constructed, but this time the program was written and verified in Coq. Only a tiny bit of code needs to be verfied in order to trust this proof (Coq's implementation of the Calculus of Inductive Constructions), and since this code is shared by all Coq users it's already had many eyes on it since appearing in the mid 1980s.

Re:Need a summary of the summary (0)

Anonymous Coward | about a year ago | (#45473271)

Four Color Theorem's tricky. The high-school definition has a known counterexample requiring only 5 regions. The college level definition excludes the nonsense that the counterexample uses.

Re:Need a summary of the summary (1)

Anonymous Coward | about a year ago | (#45473193)

You do realise that we use the difficulty of, in particular, finding large prime numbers as the basis for most modern security protocols implemented on computers? Precisely BECAUSE it's so hard to do?

We're not talking about 2, 3, 5, 7, etc. but we're talking about primes with MILLIONS of digits. Primes so large that even to prove they are prime can take a long time.

That's not how public-key crypto works at all.

Generating large primes is dead easy. This is exactly what Java keytool and OpenSSL to every time you ask them to generate a private key: they generate two large prime numbers (hundreds of digits each, not millions) and the private key is those two numbers, while the public key is the product of those two numbers. What makes this kind of crypto hard to crack is the difficulty of factoring the product of two large primes: the search space is absolutely humongous. Determining whether a given number is prime or not is much easier.

Re:Need a summary of the summary (0)

Anonymous Coward | about a year ago | (#45472805)

> Ok, I'm dense..

Prime numbers aren't dense in R. Proof is left as an excercise to the reader.

Re:Need a summary of the summary (1)

m2 (5408) | about a year ago | (#45473001)

No, they are saying that you can always find a pair of primes separated by 600. Let's say you list all the primers between 2 and N. You enumerate all the pairs whose difference is 600. What they are saying is that if you look beyond N, you will always find another such pair. They are NOT saying how much further you have to look.

They are *not* saying that given any prime number p, then p+600 is also prime.

Their goal is to demonstrate that the same is true for 2 instead of 600.

Re:Need a summary of the summary (1)

gnasher719 (869701) | about a year ago | (#45473213)

Their goal is to demonstrate that the same is true for 2 instead of 600.

The _real_ hypothesis is this: Given _any_ pattern, like (p, p+2, p+6, p+8) where it isn't obvious that only a finite number of solutions exist, there will be an infinite number of primes following that pattern.

A case where there is obviously a finite number of solutions is (p, p+4, p+8) because one of the three numbers must be divisible by 3. Or any pattern involving an odd number like (p, p + 1027); either p or p + 1027 must be even so except for p = 2 they can't be both primes.

Single mom thanked Obama (-1)

Anonymous Coward | about a year ago | (#45472525)

http://www.conservativeintel.com/2013/11/19/single-mom-thanked-obama-for-her-169mo-insurance-then-discovered-it-will-actually-cost-her-621mo/

"Jessica Sanford of Federal Way, Wash., a freelance court reporter. She isn’t just any enrollee. As it happens, President Obama once mentioned her by name. She was so thrilled at getting a “gold” level insurance plan for herself and her son for just $169 per month that she had written Obama to thank him. And then he read from her letter and gave her a name-check in his October 21 Rose Garden speech. ...
Unfortunately, Washington State did finally got back to Sanford about her application. That $452 subsidy we said you’d get? That was a mistake. You actually get zero. So for that gold plan, instead of paying $169 per month, you’d pay $621 per month."

Awww, poor little LIV.

Suck it drones. You served up a shit sandwich, now take a big ass bite and enjoy!

Re:Single mom thanked Obama (0)

I'm New Around Here (1154723) | about a year ago | (#45473261)

http://www.conservativeintel.com/2013/11/19/single-mom-thanked-obama-for-her-169mo-insurance-then-discovered-it-will-actually-cost-her-621mo/

"Jessica Sanford of Federal Way, Wash., a freelance court reporter. She isn’t just any enrollee. As it happens, President Obama once mentioned her by name. She was so thrilled at getting a “gold” level insurance plan for herself and her son for just $169 per month that she had written Obama to thank him. And then he read from her letter and gave her a name-check in his October 21 Rose Garden speech. ...
Unfortunately, Washington State did finally got back to Sanford about her application. That $452 subsidy we said you’d get? That was a mistake. You actually get zero. So for that gold plan, instead of paying $169 per month, you’d pay $621 per month."

Awww, poor little LIV.

Suck it drones. You served up a shit sandwich, now take a big ass bite and enjoy!

That has nothing to do with prime numbers, idiot.

Closing the age gap: A good companion article (0)

Anonymous Coward | about a year ago | (#45472635)

http://www.scmp.com/lifestyle/technology/article/1256542/zhang-yitang-proof-mathematicians-life-begins-40

Maybe a bit of a feel good article, but I think some deep questions are better answered by more seasoned (and likely more stubborn) mathematicians.

James Maynard 'Keenan' (0)

Anonymous Coward | about a year ago | (#45472695)

This post reminded me of the singer of the band named Tool and the song Lateralus which uses Fibonacci sequence.

http://www.youtube.com/watch?v=wS7CZIJVxFY

Still not a strong result... (1)

gnasher719 (869701) | about a year ago | (#45472727)

If N is between 2 and 10^260, then the number of primes less than N is more than N / 600. So in that range the _average_ gap between consecutive primes is less than 600. For N = 10^20 it is actually quite rare that the gap between two consecutive primes is over 600.

Re:Still not a strong result... (0)

Anonymous Coward | about a year ago | (#45472963)

Given that the ultimate goal is to look for an infinite number of prime twins, I don't think that anybody is interested in the *average* gap.

Mistake? (0)

Anonymous Coward | about a year ago | (#45472751)

FTFA:
"and if we assume the Elliott-Halberstam conjecture, that liminfn(pn+1pn) is less than or equal to 12"

But the gap between 113 and 127 is 14, so either the Elliott-Halberstam conjecture is wrong or the proof is dubious.

Re:Mistake? (0)

Anonymous Coward | about a year ago | (#45472971)

You should learn what "liminf" means.

Increasingly Rare (0)

Anonymous Coward | about a year ago | (#45472955)

So, if the prime numbers become increasingly rare as you get bigger, but never get further apart than 600, then that's a sort of interesting result for two reasons. The obvious one is that the distance between primes increases at a decreasing rate. The subtler one is that there is some magic number out that is the LEAST upper bound for this growth in distance (I sort of doubt 600 is the smallest such bound). Whatever that least upper bound is will have profound implications for the nature of the universe. If that number is a prime, is it some sort of super-prime that can be used to predict all the other primes? If that number is composite, what is special about its prime factors? Heck, that magic number might not even be an integer. It might be some sort of transcendental number like phi, e or pi. It would be really mind-blowing to know that while integers are the simplest kind of numbers that we know, they may not be the most fundamental. Euler's Identity [e^(i*pi)+1=0] has hinted at that for quite some time. We might have to rewrite the fundamental theorem of arithmetic.

Prime. Now we know. (0)

Anonymous Coward | about a year ago | (#45473107)

Nice. How soon do I get my FTL flying car?

thanks, I needed that (5, Interesting)

museumpeace (735109) | about a year ago | (#45473273)

so little of what news is dragged before me these days does much to make me hopeful of humanity's prospects on this planet. This story is the rare exception. We could be a great species. We could solve what looked for centuries to be impossible problems. We could...

Thanks /. This story was not in any of my regular channels today.
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