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## Goldbach Conjecture: Closer To Solved?170

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."
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## Goldbach Conjecture: Closer To Solved?

• #### and here is the proof for every even number (Score:4, Funny)

by Anonymous Coward on Sunday May 13, 2012 @06:52PM (#39989619)

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 (Score:5, Informative)

on Sunday May 13, 2012 @07:00PM (#39989681)

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 (Score:5, Insightful)

on Sunday May 13, 2012 @07:10PM (#39989739)

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: (Score:2)

I'm not sure what your point regarding m is, but if you just take out the 'one of these is 1' predicate it's still true since 5 primes is in the domain of 'no more than 6'.
• #### Re: (Score:3, Funny)

All y'all are confusing "theorem" with "proof". Stop it, it hurts.
• #### Re: (Score:2)

Major fail in summary - subby can't even get the conjecture right.

• #### Re: (Score:3)

Major failure in submitting at all. TT's proof was published on Febrauary 1st - this isn't news at all, this is olds.
• #### Re: (Score:3)

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

• #### Re: (Score:3)

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

• #### Re: (Score:2)

Given that 1 is divisible only by itself and 1 I hearby nominate it to be an honorary prime.
• #### Re: (Score:2)

Minor note: While that's true, in Goldbach's day it was actually common to include 1 as a prime. So the original conjectures allowed 1.
• #### Re: (Score:2)

Unfortunately (for Ramare), this was already proven before Terrence Tao's result:

We prove that every odd number N greater than 1 can be expressed as the sum of at most ve primes, improving the result of Ramare that every even natural number can be expressed as the sum of at most six primes.

• #### Re: (Score:2)

No need to make up fake primes. With 3, every even number n>4 is 3 plus an odd number n>1; 4 and 2 are, obviously, trivial.

• #### Terry Tao (Score:4, Interesting)

on Sunday May 13, 2012 @06:56PM (#39989653)
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: (Score:3)

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 (Score:5, Informative)

on Sunday May 13, 2012 @07:04PM (#39989705)

"...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: (Score:2)

Ah, yes, I knew something was missing or I disproved it inside of a couple seconds :P. Yeah, it would have to be even as all primes are odd save 2, and the sum of any 2 primes is even, so you'd be forced to use 2 as one of those primes all the time for all even number and that very quickly breaks.
• #### Re: (Score:2)

Bah, that should be "For all odd numbers". That will treat me to reed my messages before posting.
• #### Re: (Score:2, Funny)

by Anonymous Coward

That will treat me to reed my messages before posting.

Alas, it did not treat you.

• #### Re: (Score:2)

Or three primes.

• #### Re: (Score:3)

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

• #### Re: (Score:2)

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: (Score:3, Informative)

Actually... every even integer GREATER THAN TWO. See http://mathworld.wolfram.com/GoldbachConjecture.html [wolfram.com]
• #### Re:It's every *even* number (Score:5, Informative)

on Sunday May 13, 2012 @08:27PM (#39990245)

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 [wikipedia.org]. 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: (Score:2)

> For the purposes of the conjecture calling 0 prime (this is non-standard)

Calling 1 a prime is non-standard. Calling 0 a prime is er... stupid. Why ? Because it's not divisible by itself. Nothing is divisible by 0 remember.

• #### Re: (Score:2)

I did say that the cost of calling 0 prime is not worthwhile in general. Still, "nothing is divisible by 0" is a little disingenious. Compactifying the real numbers with one or two points at infinity, defining 1/0 as (positive) infinity makes some sense. Similarly defining 0/0 = 1 also makes some sense, but the usual rules of arithmetic are broken rather badly with these definitions so to avoid confusing people they're not typically made. Still, if you told a mathematician "Formula (1) is true when we inter

• #### Re: (Score:2)

Well I find that quite a fascinating proposal. X/0 is an unknown infinite (there is no such number as infinity though - we're about 200 years past that idea) but X/X = 1

So which of the two rules should actually apply to X=0 ?

Seems from your post that mathematicians pretty much interchangeably swap...

• #### Re:It's every *even* number (Score:4, Insightful)

on Monday May 14, 2012 @02:38PM (#39997591)

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

In the extended real numbers [wikipedia.org], 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 [wikipedia.org], 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: (Score:2)

2 is degenerate.

That's rather harsh, isn't it? My understanding was that 2's relationship with the goat was entirely consensual, and at worst should be deemed indecent. (Particularly the time when 2 and the goat were in front of town hall on the 4th of July with that 128 ounce jar of mayonnaise, and ... ah, but I digress.) Heck, this story talks about groups of as many as five primes getting their groove on together, but nobody's calling them degenerate. Sheesh, have a little fling with a caprid an

• #### Exhaustive search... (Score:2, Interesting)

by 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... (Score:5, Funny)

on Sunday May 13, 2012 @09:11PM (#39990505)

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: (Score:2, Funny)

by 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: (Score:2)

Sounds like it might be the perfect thing for quantum computers to handle.

• #### Re: (Score:2)

Everyone knows integers only go from 0 to 4294967295!

Hey, on my computer INT_MAX is only 2137483647. Damn store must have cheated me, giving me crippled integers. I'm going to down there right now and demand one where integers go all the way up to 0x11..11..11..11.

• #### Re:Exhaustive search... (Score:5, Funny)

on Monday May 14, 2012 @03:39AM (#39992377)

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

• #### Re: (Score:2)

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!

Can't afford a 64bit PC, huh?

• #### Re: (Score:2)

Everyone knows integers only go from 0 to 4294967295!

That's quite large [wolframalpha.com].

• #### Re: (Score:2)

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 [ams.org]

• #### Re: (Score:3)

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 (Score:2)

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 (Score:5, Informative)

on Sunday May 13, 2012 @07:18PM (#39989803) Homepage

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 [wikipedia.org] (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/ [wordpress.com]. 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: (Score:3)

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 conjectu
• #### Re:This is part of a very long trend (Score:5, Informative)

on Sunday May 13, 2012 @07:57PM (#39990025)

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 [wikipedia.org]. The statement of the Green-Tao theorem [wikipedia.org] is also accessible and interesting.

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

• #### What about negative numbers (Score:2)

When I went to school, integers included negative numbers. Of course that may have changed.

• #### Re: (Score:3)

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 do [wikipedia.org]

• #### Is this progress? (Score:5, Insightful)

on Sunday May 13, 2012 @08:22PM (#39990203)

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? (Score:5, Interesting)

on Sunday May 13, 2012 @08:38PM (#39990309) Homepage
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 [wikipedia.org] (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: (Score:3)

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).
• #### There WERE computers involved, indirectly. (Score:2)

From the abstract of Tao's paper: Our argument relies on some previous numerical work, namely the verification of Richstein of the even Goldbach conjecture up to $4 \times 10^{14}$, and the verification of van de Lune and (independently) of Wedeniwski of the Riemann hypothesis up to height $3.29 \times 10^9$.

Richstein's work (available at http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.pdf [ams.org] ) definitely involves a computer, and I assume the Riemann hypothesis verific

• #### what sort of number would prove it false? (Score:2)

Assuming Goldbach's conjecture is not true, what kind of numbers would the "anti-Goldbach" numbers be? Huge, for one. Maybe a primorial +/- 1 or +/- 9? That number at least could not be the sum of two primes in which one of the primes is a factor of the primorial. Same idea could work for a "seriously" isolated prime +/- 1, if there is such a thing. I imagine people have tried to come up with numbers that disprove the conjecture.

I'm thinking of a density and probabilistic kind of argument for Goldba

• #### What two prime numbers add together to equal 17? (Score:2)

Perhaps I didn't have my coffee this morning, or I am missing something. What two primes add together to form 17?

• #### In related news... (Score:2, Funny)

by 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.

• #### Tao's proof does rely on computer-verified results (Score:2)

Contrary to what the Fine Announcement says, and although Tao's proof itself does not require any long or involved computer calculation, it relies on previously computed results. More precisely, the proof uses a numerical bound under which the Riemann Hypothesis is known to be true. This is theorem 1.5 in his paper.

Theorem 1.5 (Numerical verification of Riemann hypothesis). Let T0 := 3.29 × 10^9.
Then all the zeroes of the Riemann zeta function in the strip {s : 0 \leq (s) \leq 1; 0 \leq
(s) \leq T0} li

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