## Mathematicians Team Up To Close the Prime Gap 194

Posted
by
Unknown Lamer

from the there-can-be-only-4680 dept.

from the there-can-be-only-4680 dept.

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."*
## Mr President (Score:5, Funny)

We cannot allow a prime gap!

## regarding collaborative efforts (Score:5, Insightful)

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 (Score:5, Funny)

Wow. That's like so deep man.

## Re: (Score:1)

## Re: (Score:3)

That's why C is always pissy.

## Re: (Score:2)

## Nice work (Score:5, Funny)

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

## Re:Nice work (Score:5, Insightful)

## Re: (Score:2)

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

## Re:Nice work (Score:5, Funny)

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

## Re: (Score:2)

thisgene causes all cancer in everyone!"## Re:Nice work (Score:5, Funny)

Seems like a good place for my favorite joke:

An Astronomer, a Physicist, and an Mathematician have traveled to England for the first time to attend a conference and are riding a train through the countryside. Before long they pass a field with a single black sheep in it. The Astronomer says, "well look at that, in England, the sheep are black." The Physicist rebukes him, saying, "how can you make such a broad statement? All we know is that in THIS field, the sheep are black." The Mathematician shakes his head in scorn at both of them and says, "gentlemen, you are both making overly general assumptions. All we can says for certain is that in England there exists at least one field, containing at least one sheep, at least one side of which is black."

## Re: (Score:2)

Proving FTW! :D

## See Kuhn (Score:4, Informative)

'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: (Score:1)

Favourite quote re said crowd taking it further:

I am not a Kuhnian.

--Thomas Kuhn

## Re:See Kuhn (Score:5, Informative)

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.

## Re: (Score:2)

We should all go re-read Kuhn anyways. Because (thinking of the current state of cosmology for one) clearly we didn't get it yet.

## Re: (Score:2)

Actually, I haven't read it yet. I only recently heard about it while reading a biography of Joseph Priestley [wikipedia.org]. I hope to read the Kuhn book soon.

Anyway, sorry if I got that wrong. I was just trying to further the always-erudite discussion here. To that end, thanks for setting me straight in the most condescending and pedantic way possible. ;-)

## Yawn. (Score:2, Funny)

>> 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 (Score:5, Informative)

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

## Re:primes separated by one (Score:5, Funny)

What do I win?

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

## Re: (Score:2)

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: (Score:2, Informative)

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? (Score:5, Informative)

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: (Score:2, Funny)

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

## Re: (Score:2)

The induction step and base case are obvious. The proof as laid out is correct for an arbitrary N. The induction step is to show that it is also true for N+1. Then your base case is to show that it is true for a specific N and N+1 like N=3 and N=4 (trivial to verify). At that point it is proven for all N where N is in the set of Natural Numbers and N >=3.

Honestly I thought it was a very well formed comment for Slashdot. You shouldn't flame something just because you don't understand it. And if you

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

This is not a proof by induction it is a proof by contradiction, no induction step is needed.

It assumes there is a number N such that their must be at 2 primes between M and M + N, for any M, then the proof goes on to show how to pick a M for which this is not the case.

unless you are referring to the proof that the numbers between N! and N! + N are divisible not primes (clearly they are since you can write it as a*k+k = a*(N + 1) where a*k=N! for all values of k between 1 and N ). But you don't need induction to prove that either.

## Re: (Score:2)

## Re: (Score:2)

The proof as laid out is correct for an arbitrary N. The induction step is to show that it is also true for N+1

Appart for the major woosh, as you didn't get the obvious joke "this holds for any N" -> "no !!! only for N-2" (and I'm not sure at this point that you will even get the hint)

You seem to have a major problem understanding the induction process which you claim to be your prefered and most intuitive way of understanding mathematical proofs. (but since you're a nice person you still admit that your GP's post is good enough for slashdot standards (thank you very much for him/her and the rest of us))

So "The

## Re: (Score:2)

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:What is the greatest lower bound? (Score:5, Funny)

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.

Wrong set. You're dealing with ALL primes. The question is about the set of KNOWN primes (you know, the ones listed in the NSA's Big Book of Primes). Between the known primes, there is a greatest known lower bound.

## Re: (Score:2)

I understood some of those words.

## Re: (Score:2)

Theoretically, it can't be any lower than 2. The fascinating thing is that as prime numbers become larger, they are found further and further apart, which plotted as a graph is more like a log n curve. But every now and again, you find a couple that are just two units apart. Usually one of them is something like (2^n)-1 and the other is (2^n)+1 . If the first one is written out as binary, it would form a prime number of 1's eg. 31. The only way such a binary number could have factors is one with 2^(n-1) nu

## Re: (Score:2)

Okay... Found them. 2 and 3.

## Re: (Score:2)

{2, 3}

## Re: (Score:1)

2, 3

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

## Re: (Score:1)

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: (Score:3)

Damn it. I got pipped to the post. And by a number of people, a number of minutes before me! I demand a recount.

## Re:Yawn. (Score:4, Funny)

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)

## Re: (Score:3)

## Factoring Primes (Score:2, Interesting)

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: (Score:1)

## Re: (Score:2)

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: (Score:2)

I submit, that there are an infinite number of quad prime sets.

[ P,P+2,P+6,P+8 that are all prime, aka, back to back pairs of twin primes]

## Re:Factoring Primes (Score:5, Informative)

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

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: (Score:2)

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.)

## Re: (Score:3)

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

Done before you posted - see upthread. Just as there's an Obfuscated C contest, Slashdot should have an "Ultimate Pedantry" contest.

## Re: (Score:3)

That was done before you posted;see up-thread. Just as there's anInternationalObfuscated CCode Contest, Slashdot should have an "Ultimate Pedantry" contest.FTFY.

## Bill Gates agrees (Score:2)

Perhaps, he was educated as to the stupidity of his remark later.

## Re: (Score:3)

So much error. So much missing.

{p,1} are the prime factors.

## Re: (Score:3, Informative)

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: (Score:2)

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

## Re: (Score:2)

## Even lower? (Score:2)

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: (Score:2)

## Problem solving abilities (Score:5, Funny)

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 (Score:5, Funny)

// [VC 2013.11.20] Fix primary oddity error in prime oddity test.#define9 015## Re:Problem solving abilities (Score:5, Funny)

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: (Score:2)

Archaic? I ran into that bug^H^H^Hnotation in JavaScript, of all things. Must be NEWWWWWWWW.

-l

/man that was an annoying bug to fix

## Re: (Score:2)

My first assembly program for the Atmel AVR gave me quite the headache debugging! The output would jump around randomly, and sometimes go backwards.

My program worked fine if I put the data tables in hex, but if I tried to put them in (what I thought was) decimal all hell broke loose. Watching the code in the emulator, I couldn't figure out why the lookups were giving me the wrong numbers.

I finally just made two data tables from 0 - 255, one in hex and one in decimal, and then programmed, and read back the

## Re: (Score:2)

## Re: (Score:2)

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: (Score:2)

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: (Score:2)

Close, but the programmer would have likely introduced a spelling error.

## Re: (Score:3)

I know, explaining a joke ruins it.

That point can't be overemphasized.

## Re: (Score:2)

Explaining a joke always makes it better.

## Summary (Score:2)

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: (Score:1)

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

## Re: Summary (Score:1)

## Re: (Score:3)

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

wasYitang Zhang who settled the original long-standing open question - that being, is thereanynumber 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

## Re: (Score:2)

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

Err, ignore this. Getting confused.

## Re: (Score:2)

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

The problem is you're not getting it. The unproved, but widely believed to be true, conjecture is: you will

alwaysbe able to find, as you count higher and higher, more pairs of primes separated by 2.## Re:Summary (Score:5, Insightful)

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 (Score:5, Insightful)

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.

## Re: (Score:2)

You're driving on a highway leaving a city. At every prime numbered mile marker there's a gas station. As you leave the city the gas stations are close together, with a station at the 2 mile marker, another at the 3 mile marker, another at the 5 mile marker, etc. As you get into the suburbs the gas stations are less frequent. As you get into the desert you find that gas stations are hard to find.

But you notice something - it seems that no matter how far you drive into the de

## Twin primes ? (Score:3)

## Re: (Score:2)

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

## Still not a strong result... (Score:2)

## Re: (Score:2)

## thanks, I needed that (Score:5, Interesting)

Thanks

## Re: (Score:2)

True, stories like this, and those related to my crack-smoking mayor, give me a reason to get up each morning.

## The 600 gap: Good news, everybody! (Score:2)

You've got to find them, though...

## Where do such large numbers come from? (Score:3)

The linked abstracts are pretty vague. Are there any mathematicians here who can explain how (seemingly arbitrary) large numbers like 600 or 70 million come out of these proofs? People are saying they're all tweaks of the same basic method, so what is that basic method, exactly?

## My modified Goldbach Conjecture (Score:2)

Note if you can find the proof of this, then you have killed multiple birds with one stone.

You get the infiinite twins problem solved.

You get the Goldback conjecture solved.

And you find that is also shows that th

## Out of curiosity... (Score:2)

## Wait a minute... (Score:2)

## Re: (Score:3)

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

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: (Score:2)

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: (Score:2)

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 (Score:5, Informative)

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 (Score:5, Informative)

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 (Score:5, Informative)

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

notthat 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 (Score:4, Informative)

all the numbers between them are composites.

Ahem. Those are the two largest

knownprimes (because primes of that form are particularly easy to search for using existing techniques), but there's nothing to say that there are not unknown primes between them. In fact, there almost certainly are many; the density of primes in that region should be on the order of 1 in every 100 million integers, so there are probably at least about 10^17425161 other primes in that span.## Re:Need a summary of the summary (Score:5, Informative)

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 (Score:5, Informative)

No, that's not the theory at all. The theory does not say there is always a prime within 600 of another (that's simply not true).

The theory says for

anynumber X, there is a pair of primes larger than X within 600 of each other. That pair may be 2 larger than X, 12 larger than X, or 21,515,359 larger than X.Everything else you said is pretty much spot on though.

## Re: (Score:2)

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: (Score:2)

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.

## Re: (Score:2)

No.

What this theory says is that no matter how far up you look on the number scale, you can always find a pair of larger primes that are separated by less than 600.

i.e. for any number X you always find primes larger than X that are closer than 600 from each other

In the opposite direction (what is the maximum gap between primes), the gap increases without bound.

i.e

For any number X you can always find closest primes that are more than X apart.

Here's a proof:

Take any number N

N! = (N) x (N-1) x (N-2) x...x (3)

## Re: (Score:2)

You're driving on a highway leaving a city. At every prime numbered mile marker there's a gas station. As you leave the city the gas stations are close together, with a station at the 2 mile marker, another at the 3 mile marker, another at the 5 mile marker, etc. As you get into the suburbs the gas stations are less frequent. As you get into the desert you find that gas stations are hard to find.

But you notice something - it seems that no matter how far you drive into th