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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."
I hereby prove that every even number is a sum of no more than six primes, one of those is 1.
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!
Really.
"...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.
-- Insert witty one-liner here. --
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.
Think of all of the applications this can be used for...
Ok ... while I'm having difficulty at the moment coming up with one that's probably because of the electrodes they attached to my head to make me happy.
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".
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?
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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.
Optimus and Giedi?
What about the integer of ZERO? What two primes represent that, noting that primes are >1 (ie. not ZERO and not negative). I'm no mathematician and this occurred to me in about 5 seconds, so I'm sure it's flawed thought from me.
AC
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]
Ask me about repetitive DNA
When I went to school, integers included negative numbers. Of course that may have changed.
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.
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
"this is classical mathematical proof involving logical deductions rather than exhaustive search"
I would be very impressed with an exhaustive search covering every even integer.
The proof is missing a crucial part at the end...
Q . E . D .
There!
"no computers where involved in the proof"
should be
"no computers were involved in the proof"
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 ) definitely involves a computer, and I assume the Riemann hypothesis verification does as well.
Still better than Sentinel Prime.
Computers were used in the proof! An exhaustive search was done over some small cases; and there have been previous proofs for the integers that are "large enough". Tao managed to make large enough small enough so that it falls into the rang covered by the computerized exhaustive search,
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 integers which are constructed from the sum of two odd numbers.
So is the article wrong? Should Goldbach actually be limited to 'every even integer' or does the mistake lie somewhere else?
Watch the movie "Fermat's Room", which is a great movie.
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)
My first Journal Entry ever, in 8 years! http://slashdot.org/journal/365947/aphelion-scifi-fantasy-horror-poetry-webzine
All primes except 2 are odd.
Otherwise 2 itself would be a factor which would make it not so prime anymore.
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 Goldbach's conjecture. There are approximately x/ln(x) primes less than x. The chance that any two random numbers less than x is prime is (1/ln(x))^2. The probability that none of the pairs of odd numbers that sum to the even number in question are prime is (1-(1/ln(x))^2)^(x/4). As x gets larger, this quantity shrinks. In other words, the likelihood that an even number will have 2 primes that sum to it goes up as the number gets larger.
Has anyone counted how many ways there are to satisfy the conjecture for particular numbers? Are there more pairs of primes that sum to a particular even number, when that number is larger? There any big even numbers for which there is only one pair of primes that sum to them?
I also thought of trying for an even stronger conjecture. How could Goldbach's be made stronger? How about requiring that every even number x be the sum of not just 1 pair of primes, but the sum of ln(ln(x)) or sqrt(ln(x)) distinct pairs of primes? I'm just tossing those formulas out there, no idea if they're reasonable.
Intellectual Property is a monopolistic, selfish, and defective concept. It is "tyranny over the mind of man"
Oops. Sum of 3 (prime) and 4 (NOT prime). Maybe in the summary he meant "sum of primes" not "sum of TWO primes" (emphasis added to enhance flavor). But I just disproved the conjecture right there, if the conjecture actually is that every integer is the sum of TWO primes. (That, again, is unless you mean sum of any multiple of two different primes, in which case 2 (prime) plus 2 (prime) plus 3 (prime) IS seven... I would go ahead and chalk this up to "bad summary".
Perhaps I didn't have my coffee this morning, or I am missing something. What two primes add together to form 17?
Aha - I think the summary is perhaps misleading. Wikipedia explains its only even integers: http://en.wikipedia.org/wiki/Goldbach's_conjecture
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.
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.
'every integer can be expressed as the sum of two primes'
were correct, Goldbach's conjecture would be a very simple way to find a huge number of primes.
If so, any given odd integer would be the sum of an even prime (with a very limited choice, namely the number 2) and another prime, one of the odd ones.
If so, any given odd integer would lead to a prime, the ultra-easy way.
But it doesn't. You don't necessarily get a prime as a result by just taking an arbitrary odd integer and subtracting 2 from it. In most cases, you'll get another odd integer and that's it.
So 'every even integer can be expressed as the sum of two primes' is a more precise way to put Goldbach's conjecture.