## Goldbach Conjecture: Closer To Solved? 170 170

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

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)

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)

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(wheren> 4) is a sum of no more than six primes, becausem=n- 3 is an odd number".## Re: (Score:2)

## Re: (Score:3, Funny)

## Re: (Score:2)

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

## Re: (Score:3)

## 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.

## The numbers less than 3 (Score:4, Insightful)

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: (Score:3)

of course, Goldbach was a bit before Ring theory [wikipedia.org], 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 (Score:4, Informative)

## Re: (Score:2)

Correct.

Negative numbers are not prime.

1 is not prime (though it never fucking matters since it's always the trivial/base case when you're doing anything useful).

## Dis-proof of Goldbach as stated? (Score:2)

every integer can be expressed as the sum of two primes

but this seems trivially easy to disprove. There is only one even prime, 2, so if I take an odd integer I have to construct it from the sum of an even and an odd number hence if N-2 is not a prime number Goldbach (as stated) cannot be correct. Now consider '11': since 9 is not a prime number and '2' is the only even prime this cannot hold true for all integers, only even in

## Re:Dis-proof of Goldbach as stated? (Score:5, Informative)

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

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.

## Slashdot Summaries Again (Score:5, Informative)

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 [utm.edu]

http://primes.utm.edu/glossary/xpage/OddGoldbachConjecture.html [utm.edu]

(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: (Score:2)

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.

By the time I read it, using my Slashdot settings it was the second post I read. The moderation system mostly works.

So for a community that is expert on Forks, why can't we just Fork Slashdot?

Wasn't that essentionally what Technocrat was?

The only value they offer is the "summaries" and *every single one is wrong*.

Oh really?

We lost our leader anyway

Speak for yourself.

## Re: (Score:3)

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: (Score:2)

I think the Slashdot effect might have something to do. I don't know how bad it is lately, though; maybe a small farm of modern servers may stand the load - but who would pay for it?

## Re: (Score:2)

By the time I read it, using my Slashdot settings it was the second post I read. The moderation system mostly works.

Say something nice about the mod system and get the old additive identity mod.

So for a community that is expert on Forks, why can't we just Fork Slashdot?

You could call it \. and aid confusion about slashes.

We lost our leader anyway

Maybe he got tired of a certain comment.

Speak for yourself.

Are you sure that's ok?

## Re: (Score:2)

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## Re: (Score:2)

As others have noted, it depends. It's sometimes convenient to consider -1 as being prime, for example, because it allows you to extend the notion of squarefree numbers to negative integers.

## Re: (Score:2)

## Re: (Score:2)

You may have missed the word "squarefree" in my comment.

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## 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)

## 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)

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

It should be every

eveninteger. Note TFA has sums for 52, 54, 56, 58 and 60.## Re: (Score:2)

## Re: (Score:2)

## Re: (Score:2, Funny)

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 threeis halfway between two primes."Your sig is confusingly appropriate ;-)

## Re: (Score:3, Informative)

## Re:It's every *even* number (Score:5, Informative)

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)

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)

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

themdegenerate. Sheesh, have a little fling with a caprid an## Re: (Score:2)

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

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)

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)

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)

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)

Computers

wereinvolved 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)

## 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)

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)

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)

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.

## Re: (Score:2)

Hah. I actually am gay, but nope, I don't have a man-to-man crush on him, just his math. I can't imagine how exhausting it would be to try keeping up with him mentally, which makes me curious about his wife.

## 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]

## Re: (Score:2)

Thank you, that is a good point. I neglected the overall sign on the factorization. -1 would then be -{empty product}.

## Re: (Score:2)

I'm sorry, but you have no idea what you're talking about. I suspect you're trolling, but if not see this definition of prime element [wikipedia.org]:

In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b

-2 is prime by this definition since it's not zero, it has no multiplicative inverse in the integers, and whenever ab is divisible by -2, either a or b is divisible by -2.

This definition can be used in the defining property of unique factorization domains (see my original link) to generalize the fundamental theorem of arithmetic to other rings. Strictly speaking irreducibles

## Re: (Score:2)

Words can have multiple definitions, even in math. The common definition of "prime" refers to positives, but the definition I'm referring to (as made more explicit here [wikipedia.org]) allows negatives. I was explicit about which definition I was referring to ("the 'prime numbers' used in abstract algebra"), so my usage was correct.

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

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

allexponents 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)

## Re: (Score:3)

## 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?

## Re: (Score:2)

Aha - I think the summary is perhaps misleading. Wikipedia explains its only even integers: http://en.wikipedia.org/wiki/Goldbach's_conjecture [wikipedia.org]

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

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.

## Re:Every Integer? (Score:5, Insightful)

7 + 2 = 9

## Re:Every Integer? (Score:5, Funny)

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

## Re: (Score:3)

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: (Score:2)

Brief note of course is that you cannot do an exhaustive search on numbers since by definition they are infinite. This is no finite set therefore could not be exhaustively searched, why would anyone think it could is beyond me.

## Re: (Score:3)

The continuous plane is infinite too, yet the seminal math proof using computers was the four-color map theorem [wikipedia.org], 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: (Score:2)

I think he gets it; just won't admit it. At least you understood my point.

## Re: (Score:3, Funny)

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: (Score:2)

Sorry I mean 11 can't be, because one of the terms would have to be 9 which isn't prime.

## Re:Every Integer? (Score:4, Funny)

7 + 2 + 2

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

## Re:Every Integer? (Score:4, Informative)

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: (Score:2)

True, and that would meet the sum of three odd primes requirement. If you include 2, then 7+2=9, making it the sum of two primes where one prime is NOT a Goldbach Number.

ObTrivia: Found this page, it looks like it has some interesting information: http://homepage.mac.com/billtomlinson/primes.html [mac.com]

## Re: (Score:2)

http://mathworld.wolfram.com/GoldbachNumber.html [wolfram.com]

Wolfram says the same thing. So Nyah! Pttttthhhhhpt!

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## Re: (Score:2)

What?

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

## Re: (Score:3)

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:

## Re: (Score:2)

Why can't submitters get it through their head that when you quote someone, you need to put the quote in quotation marks and give credit to the source?That's quite a leap of faith that the submitter got it wrong...

## Re: (Score:3, Informative)

eveninteger greater than 2 can be expressed as the sum of two primes." (source [wikipedia.org])## Re:ZERO? (Score:5, Funny)

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

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## Re: (Score:2)

If you have Maple handy, this one-liner does the trick:

a := whatever; b := 2; while not(isprime(a-b)) do b := nextprime(b) end do; a-b, b;Of course the interesting part is the code for isprime(). You can start here [wikipedia.org] if you want to know how it's done.

## Re: (Score:2)

I used the GMP [gmplib.org]. It includes an isprime() function. I've been meaning to install it on my computer, and the assertion by sexconker gave me the motivation to actually do it. It just took a simple decrement by two and test for primality loop.

## Re: (Score:2)

I suspect it would have been disproved a long time ago.

Long, long ago. If the conjecture really was as written, it would require every odd integer to be prime, which is patently ridiculous (eg. 3^k would be prime). Suppose the number n is odd. n+2 is odd too and, if the misstated conjecture were true, n+2 = p+q for primes p and q. Since all primes except 2 are odd, to get the sum to be odd, one of p or q must be 2, say q=2. But then n+2 = p+2, so n = q, and n is prime since q was.

## Re: (Score:2)

All primes except 2 are odd.

Otherwise 2 itself would be a factor which would make it not so prime anymore.

## Re: (Score:2)

Optimus and Giedi?

Optimus and Rodimus. Deal with it.

## Re: (Score:2)

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." [wikipedia.org], so I agree the summary isn't accurate. =)