Goldbach Conjecture: Closer To Solved?

mikejuk writes "The Goldbach conjecture is not the sort of thing that relates to practical applications, but they used to say the same thing about electricity. The Goldbach conjecture is reasonably well known: every integer can be expressed as the sum of two primes. Very easy to state, but it seems very difficult to prove. Terence Tao, a Fields medalist, has published a paper that proves that every odd number greater than 1 is the sum of at most five primes. This may not sound like much of an advance, but notice that there is no stipulation for the integer to be greater than some bound. This is a complete proof of a slightly lesser conjecture, and might point the way to getting the number of primes needed down from at most five to at most 2. Notice that no computers were involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search."

Comments

and here is the proof for every even number

[Anonymous Coward]

I hereby prove that every even number is a sum of no more than six primes, one of those is 1.

Re:and here is the proof for every even number

[santiago]

I hereby prove that every even number is a sum of no more than six primes, one of those is 1.

Psst, 1 isn't prime. Or composite. It's neither.

Re:and here is the proof for every even number

[Smurf]

I hereby prove that every even number is a sum of no more than six primes, one of those is 1.

Psst, 1 isn't prime. Or composite. It's neither.

True, but you can change the GP's proof to "every even number n (where n > 4) is a sum of no more than six primes, because m = n - 3 is an odd number".

Re:and here is the proof for every even number

[Geoffrey.landis]

Every even number greater than 1 is the sum of no more than six primes, one of which is three.

Re:and here is the proof for every even number

[Sussurros]

Given that 1 is divisible only by itself and 1 I hearby nominate it to be an honorary prime.

Re:and here is the proof for every even number

[silentcoder]

What about 2 ? It's even, and it's greater than 1, but it's less than your 3.

Re:and here is the proof for every even number

[Garridan]

All y'all are confusing "theorem" with "proof". Stop it, it hurts.

Re:and here is the proof for every even number

[fatphil]

Major failure in submitting at all. TT's proof was published on Febrauary 1st - this isn't news at all, this is olds.

Terry Tao

[bgeezus]

Terry Tao always amazes me with the scope of his knowledge. Contributions in mathematical areas as diverse as random matrix theory, harmonic analysis, and number theory. I look forward to what comes next!

Re:Terry Tao

[EuclideanSilence]

I've also heard Tao say in lecture that he doesn't even like using computer assistance when he's working out theory. I found some of his lectures to be great for getting the scope of ideas, but unless you really know the subject of number theory he can be hard to follow.

It's every *even* number

[MrKevvy]

"...every integer can be expressed as the sum of two primes."

It should be every even integer. Note TFA has sums for 52, 54, 56, 58 and 60.

Re:It's every *even* number

[cupantae]

Another way to say it, which just occurred to me now, is:
"Every natural number is halfway between two primes."

Re:It's every *even* number

[su5so10]

Actually... every even integer GREATER THAN TWO. See http://mathworld.wolfram.com/GoldbachConjecture.html

Re:It's every *even* number

[FrootLoops]

Indeed, it should actually say, "every even integer greater than 2 can be expressed as the sum of two primes". 2 is degenerate. For the purposes of the conjecture calling 0 prime (this is non-standard) gets rid of that little wrinkle, though the cost of a more involved statement of the fundamental theorem of arithmetic is not worth it (which is incidentally a good reason why 1 isn't prime).

For anyone interested, an actual theorem that's similar to the Goldbach conjecture is Lagrange's four-square theorem. It states that any non-negative whole number is the sum of the squares of four whole numbers. There are numerous proofs, though I wouldn't recommend trying to find one yourself if you don't have a background in algebra or number theory.

Re:It's every *even* number

[Anonymous Coward]

That will treat me to reed my messages before posting.

Alas, it did not treat you.

Re:It's every *even* number

[FrootLoops]

Actually "infinity" is an honest number in several modern, rigorous senses.

In the extended real numbers, one adds two symbols to the usual real numbers (which won't render here), "+inf" and "-inf". No mystical qualities are needed; one could just as well use symbols "@" and "#". The extended real numbers are useful in formulating elementary measure theory, where some basic arithmetic with them is defined (+inf - -inf = +inf, for instance; +inf + -inf is left undefined).

In the real projective line, one adds a single "point at infinity" which is imagined to "wrap around" from "negative infinity" to "positive infinity". I'm sorry for all the scare quotes; the actual construction is rigorous. Suppose you have a plane and a horizontal line passing through y=1. Given a point on the horizontal line, there is another line passing through that point and the origin; this line is taken to be a "point" on the real projective line. The additional point at infinity is taken to be the horizontal line passing through the origin, which is the limiting value of the other real projective line-points as they go to positive or negative infinity.

As for X/X = 1 vs. X/0 = infinity when X=0, one could simply say X/0 = infinity when X is not 0 and then there is no conflict. But again, the usual rules of arithmetic don't work well in this situation, so you need a good reason to extend arithmetic to work with infinities. The only case I've encountered where that is true is with the extended real numbers in measure theory mentioned above.

As for mathematicians, yes, we change conventions whenever needed without real difficulty. The phrase "ring" is a great example--it can have a huge variety of meanings depending on context. Careful authors will specify, but otherwise you'll have to figure out from context what precisely is meant. Once in a while this can be confusing, but for something as simple as whether primes can be negative or not it's a complete non-issue.

Re:It's every *even* number

[Anynomous Coward]

Nicely stated, but not correct unless you consider 1 to be prime, which is as much blasphemy as stating that Pluto is a planet.

Try "Every natural number above three is halfway between two primes."

Your sig is confusingly appropriate ;-)

Re:Every Integer?

[jejones]

7 + 2 = 9

Exhaustive search...

[Anonymous Coward]

Notice that no computers where involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search.

Exhaustive search for a result that holds for every integer? Good luck with that one.

Re:Exhaustive search...

[sexconker]

Notice that no computers where involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search.

Exhaustive search for a result that holds for every integer? Good luck with that one.

Everyone knows integers only go from 0 to 4294967295!

Re:Exhaustive search...

[Anonymous Coward]

They recently discovered a few more: 4294967296 through 18446744073709551615. Just in time too--we were starting to run out in some computations. Unfortunately it'll take a bit longer to verify the conjecture for these newly discovered specimens. At least there's only a finite number of primes...

Re:Exhaustive search...

[silentcoder]

You youngsters... I remember telling that joke with 32768.

Re:Exhaustive search...

[doru]

Notice that no computers where involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search.

Computers were involved to some extent. From Tao's blog:

The first refinement, which is only available in the five primes case, is to take advantage of the numerical verification of the even Goldbach conjecture up to some large {N_0} (we take {N_0=4\times 10^{14}}, using a verification of Richstein [...])

. See the paper by Richstein: http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.pdf

Re:Exhaustive search...

[martas]

You do know that there are an infinite number of planar graphs, yes? You do also know that the (first) proof of the 4-coloring problem involved exhaustive search over a finite number of sub-problems, yes? So what's your point? (If it was just a joke, then my apologies)

Exhaustive

[lurker1997]

every odd number greater than 1 is the sum of at most five primes

Notice that no computers where involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search."

That would have been a pretty long "exhaustive search".

This is part of a very long trend

[JoshuaZ]

Work on this problem has been ongoing for about a hundred years now. First, Schnirelmann proved that there was some k such that every even integer could be expressed as a sum of at most k primes. The value for k had then been reduced over time. Vinogradov's proved that the Odd Golbach Conjecture (that every odd integer greater than 7 is the sum of three primes) was true for sufficiently large n. How large sufficiently large is has been slowly reduced. Later in the 1970s, Chen proved that every sufficiently large even integer is the sum of a number that is prime and another number that is either prime or a product of two primes. At this point, Chen's result is the strongest result known.

In general, there are two general methods of attack on this problem, one which uses Schinerlmann's method and variants thereof, and the other which uses sieve theoretic approaches with the Hardy-Littlewood circle method http://en.wikipedia.org/wiki/Hardy-Littlewood_circle_method (Chen used a version of this for his result and Tao's work uses a similar approach). Unfortunately, there's not much work on actually connecting the two methods. There's an excellent piece of Tao at his blog where he discusses his work on the problem and is understandable without much background. http://terrytao.wordpress.com/2012/02/01/every-odd-integer-larger-than-1-is-the-sum-of-at-most-five-primes/. Note that TFA is a little out of date since he announced this result with a preprint a few months ago, and it is only that now the result is being published.

It does not seem that this result really does put us much closer to proving the full Golbach Conjecture. At most this could be used to prove some version of the odd Goldbach Conjecture. The methods used would have a large amount of trouble dropping from 5 to 3. There's some bit of leeway, and if anyone is going to do it, it is going to to be Tao, but right now, I'm not optimistic.

Re:This is part of a very long trend

[gnasher719]

It is astonishing how weak the result is, and how hard it is to prove.

The "ordinary" Goldbach conjecture is: Every even number N >= 4 is the sum of two primes. For example, 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53, so we see that sum numbers can be written as the sum of two primes in many different ways. We call a number that is the sum of two primes a "Goldbach number", then the conjecture says that every even integer N >= 4 is a Goldbach number.

The "weak" Goldbach conjecture is: Every odd number N >= 7 is the sum of three primes. At least one of these primes nust be odd, so we can express the weak conjecture as: For every odd N >= 7, there is an odd prime p = N - 4 such that N - p is a Goldbach number. Since there are many odd primes p = N - 4, only one out of a huge list of numbers of the form N - p need to be a Goldbach number.

Re:This is part of a very long trend

[FrootLoops]

and if anyone is going to do it, it is going to to be Tao, but right now, I'm not optimistic.

Agreed. I imagine Terry Tao isn't well-known outside of mathematics, but for those who don't know, he's certainly one of the most famous and skilled living mathematicians. He's originally Australian and is currently at UCLA. His list of high profile awards is ridiculously long, but aside from top-notch research, he's also an excellent teacher. His blog is mainly pitched at math grad students and higher, but some of it is very accessible. There's of course more biographical details at his Wikipedia page. The statement of the Green-Tao theorem is also accessible and interesting.

I totally have a researcher-crush on him, or more specifically his math skills.

Re:Every Integer?

[Old Wolf]

Wow, what has slashdot come to when posts are getting modded up for posting basic arithmetic :)

Re:ZERO?

[Spodi]

Goldbach's conjecture: "Every even integer greater than 2 can be expressed as the sum of two primes." (source)

The numbers less than 3

[gringer]

If you're talking about integers (which this conjecture refers to), then that's easy:

2 = 5 + -3

0 is trivial:

0 = p + -p for all prime numbers p

1 is also fairly easy:

1 = 3 + -2

And just to complete this, here's 3:

3 = 5 + -2

[multiplication by a unit, in this case -1, does not change the "primeness" of a number]

Re:The numbers less than 3

[gringer]

of course, Goldbach was a bit before Ring theory, so may not have been referring strictly to "todays" integers, or prime elements in the set of integers (i.e. including negative numbers).

Re:The numbers less than 3

[mark-t]

fwiw, it's my understanding that negative numbers are not considered primes, since allowing primes to be negative would allow composite numbers to have non-unique prime factorization.

Re:Every Integer?

[ChrisMaple]

29-2

Is this progress?

[spaceyhackerlady]

Sorry, but I can't accept this being progress toward a proof.

Consider Fermat's Last Theorem. Proving it for any particular exponent is doable. Mathematicians had proved it for various sets of exponents (Sophie Germain, Wieferich, etc.). But the proof for all exponents was based on completely different mathematics (Elliptic curves/modular forms, Taniyama-Shimura, Wiles) and didn't look like anything that had come before.

...laura

Re:Is this progress?

[JoshuaZ]

That's not completely true. Wiles's proof only proves it for an exponent that is a prime p>=7. So one needs the classical results of n=3,4,5,7 also. This is to some extent a minor criticism. Your essential point is correct that sometimes a proof of a theorem comes out of a completely different direction. But, very often, it does come from a straightforward way of refining the same techniques more and more. For example, Catalan's Conjecture http://en.wikipedia.org/wiki/Catalan's_conjecture (the claim that that the only consecutive positive perfect powers are 8 and 9) was proven by what in many ways amounted to slow and steady progress.

Re:Is this progress?

[phantomfive]

Even in the worst case, and the ultimate proof doesn't look anything like this, he's still eliminated one path that someone else might try (or forged along that path to show what it could do).

Re:What about negative numbers

[FrootLoops]

Integers do still include negatives. Actually, the "prime numbers" used in abstract algebra also include negatives, so for instance -5 is prime. This genuinely useful convention results in the following statement of the fundamental theorem of algebra: "Every non-zero integer can be factored as the product of prime numbers. The factorization is unique up to order and signs." (Example: -12 = 2*(-2)*3 = (-2)*(-3)*(-2).) This directly generalizes to a corresponding statement in so-called unique factorization domains, of which the integers are a particular case. Still, fiddling with negatives might confuse students, and people don't often factor negative numbers anyway, so the definition is often restricted to positive numbers, even though it's artificial.

Re:Every Integer?

[Burpmaster]

7 + 2 = 9

Damn, that's the most intelligent post I've seen on Slashdot all day, and I mis-clicked and chose 'redundant' when moderating...

Re:Every Integer?

[sexconker]

7 + 2 + 2

Ah, Mexican Math, we meet again. That's not two primes. That's three primes, two of which are 2.

Re:Dis-proof of Goldbach as stated?

[rpresser]

http://mathworld.wolfram.com/GoldbachConjecture.html

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers >=4 can be expressed as the sum of two primes.

Re:Every Integer?

[jd]

Looked the conjecture up on Wikipedia. It's actually a little more specific still - every even number is a Goldbach Number, where a Goldbach number is a number that can be written as the sum of two odd primes.

That means that every odd number can always be written as the sum of three primes or less. Numbers like 9 are the sum of two primes but are NOT Goldbach numbers since one of the primes is 2 and the requirement is that both primes be odd.

Errors in this post are due to Wikipedia, blame them if there are any.

Re:and yet another plagiarized slashdot summary

[bcrowell]

What?
The entire summary is quoted (error and all) from the only linked article. How is that not giving credit?

It's not giving credit because it says: 'mikejuk writes "...",' where ... is a collection of sentences grabbed from various places in the article, and none of those sentences were written by mikejuk.

Similarly --

bcrowell writes:

O Romeo, Romeo! wherefore art thou Romeo? Deny thy father and refuse thy name. Parting is such sweet sorrow, that I shall say good night till it be morrow. Courage, man; the hurt cannot be much.

Slashdot Summaries Again

[TaoPhoenix]

Dammit, Slashdot you have some of the best commenters here but you're wasting our time making us get about 30 comments in before someone posts the correction to the flawed summaries.

From what I can see in a quick glance, the summary is at least partially wrong. The "regular" Goldbach conjecture seems to apply to every *even* integer greater than 2. So your odd number question disappears into another heading, which is apparently called variously the odd-number or three-primes version of the Goldbach.
http://primes.utm.edu/glossary/page.php?sort=goldbachconjecture
http://primes.utm.edu/glossary/xpage/OddGoldbachConjecture.html

(Rant)

So for a community that is expert on Forks, why can't we just Fork Slashdot? *We* are the "value". The only value they offer is the "summaries" and *every single one is wrong*. We lost our leader anyway, and we've all seen what the successors are up to, and Slashcode is sorta/mostly open source right? (Dunno if they bolted on something.)

So why can't we Fork Slashdot? Are we so exhausted and burnt out from the days when fighting IE6 and Vista mattered, that we just don't care anymore? Oh and by the way, every new user would start at the *bottom* of the thread so those new breeds of shills with names like SunriseVista and BoldBraveBalmer don't hijack the top real estate of the conversation. P.S. Sorry, AC's, the top 10 memes of 2003 Slashdot have to go to now. Basically no other forum on the entire net has the First Post thing, and while I get the low level "test against censorship thing", we need a *user option* to flip the entire first post thread and any matching titles to the *bottom* of the post set. Then the *second thread in* which tries to deal with the article can do some work.

(/Rant)

Re:Slashdot Summaries Again

[DarkIye]

I have a feeling you're basically describing Reddit. I have no doubt that there's an /r/maths, and it's probably quite good...

Ugh. Fine, /r/math, no 's'. Yeah, it doesn't look bad.

Re:ZERO?

[WalksOnDirt]

Not so, 8561290356012956901265912656135612056135460123560912356102650931951 and 653 are prime. They sum to your number.

Re:Every Integer?

[Raenex]

a summary that was quite excited to point out that computation isn't the same as proof

It doesn't say that. It says, "Notice that no computers were involved in the proof -- this is classical mathematical proof involving logical deductions rather than exhaustive search."

If you're going to bitch, at least complain about the incorrect statement of the theorem.

Re:Every Integer?

[Raenex]

The continuous plane is infinite too, yet the seminal math proof using computers was the four-color map theorem, which sparked a controversy that continues to this day:

"Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property."

Re:Seven.

[Roujo]

How about, say, 2 + 5?

Mind you, you'll run into a problem when you get to 11. As stated elsewhere in the comments, though, the Conjecture actually says "Every even integer greater than 2 can be expressed as the sum of two primes.", so I agree the summary isn't accurate. =)

In related news...

[Anonymous Coward]

I'm going to eat 5 donuts a day while masturbating to pictures of Angela Merkel. It's not the sort of thing that relates to practical applications, but they used to say the same thing about electricity.