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Grok Goldbach, Grab Gold
Caseman writes, "Are you a closet mathematician who wants to come out? British publisher Tony Faber is offering a cool million bucks to the first would-be math head to prove the infamous Goldbach conjecture. Yeah, the one about every even number being the sum of two primes. Impossible, you say? Remember Fermat's Last Theorem? It stood unproven for 350 years until a recent effort yielded a proof. Read The London Times article about the challenge. "
What if you disprove the theorem? Then what, huh? No money?
1 + 3 + ... + n = (n + 1)^2 / 4
We know that, for n = 5, 1 + 3 + 5 = (5 + 1)^2 / 4 = 9, and so the formula is true for n = 5. Now assume that the formula is correct for all n. What happens if we work it for n + 2?
1 + 3 + ... + n + (n + 2) = ((n + 2) + 1)^2 / 4 - substitute n+2 for n in the formulas ... + n
(n + 1)^2 / 4 + (n + 2) = ((n + 2) + 1)^2 / 4 - substitute the assumed true value in for 1 + 3
(n + 1)^2 / 4 + (n + 2) = (n + 3)^2 / 4 - add n + 2 and 1
(n + 1)^2 + 4(n + 2) = (n + 3)^2 - Multiply through by 4
(n^2 + 2n + 1) + (4n + 8) = n^2 + 6n + 9 - multiply everything out
n^2 + 6n + 9 = n^2 + 6n + 9 - add everything up. This is true for all n.
What all that means is that if the formula holds for n, then it must hold for n + 2. And since it holds for n + 2, it must hold for n + 2 + 2, and so on into infinity. I came up with one n (5) for which the formula works, so it must work for all odd numbers larger than five. I don't have to test all the odd numbers into infinity - the infinitely recursive nature of induction does that for me in something that can be expressed in a noninfinite amount of data (and a good thing, too - I don't have enough bandwidth to post an infinite-length comment to /.)
This, in turn, means that for every integer I, there is a circle, such that the centre is at I and the intersection of the circle with the positive natural numbers will occur at two prime numbers.
Now, how does this make things any easier? Simple. If the distribution of primes is =totally= random, this statement HAS to be false. Randomness means that by knowing any number of points, you cannot deduce an unknown point. Here, we have shown that by knowing one point and a centre, we can deduce another point. This implies an inherent relationship between the prime numbers.
Thus, IF AND ONLY IF the prime numbers are non-random, Goldbach's Theorum is correct. If, however, they ARE random, it would be impossible to define a geometric relationship, which (as shown) does exist.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
You are holding waaay too much in memory. If you have 7000 primes, and you consider all pair of them, that is 49,000,000 numbers. If you were slightly smart about it you could cut that in half.
As you noticed, this is pretty inefficient.
A much better way to slice this problem is to look at the numbers as an array of bits. For instance the first 800,000 numbers is a 100 K string. Start off with a block of 0's. Generate a vector of primes. Shift the vector by 2. OR it with your first vector. Shift by 1 more. OR it with your first vector. Shift it by 2 more. OR it again. Shift it by 2 again, then 4, then 2... (We are walking through cumulative shifts equivalent to the number of primes.)
You can make it faster still by having your bitmap being only the even numbers. You will have to do a bit of thinking. But you should find that this approach is significantly faster and cuts down tremendously on the memory. You will also find that after a few shifts you will have covered most of the territory. At some point it becomes more efficient to track down each one you have not tracked down and search for primes that add up to that one...
Cheers,
Ben
My usual seat in the cluetrain is at A HREF="http://pub4.ezboard.com/biwethey.ht
With the probability reasoning it is trivial that there are (100% probability) only a finite number of exceptions. The odds of there being any after you have verified the first 100,000 cases is pretty small. But it gives you no idea how to prove it.
:-/
Likewise the probability result strongly suggests that the Riemann conjecture is true. But again, it gives no insight into the real problem.
However when you compute statistics, perfect match with the probability prediction.
Cheers,
Ben
My usual seat in the cluetrain is at A HREF="http://pub4.ezboard.com/biwethey.ht
That's definitely not the only way to prove stuff like this. It goes back to the fact that you can use a generalized definition to talk about all those numbers up to infinity, and then prove based on the definition instead of on real numbers.
:)
For example, prove that every single possible even number is the sum of two odd numbers. Instead of looking at all the examples, we'll go by definition.
Definition of even number: 2x, where x is an integer.
Definition of odd number: 2y-1, where y is an integer.
So if the even number is 2x, that is the same as 2x-1 + 1, which is two odd numbers added together.
That was a really simply example, but hopefully you get the idea. No matter what even number you choose, it fits that definition. 993928302390 fits the definition, and plugging it into the equation, the two odd numbers are 993928302389 and 1. I just didn't have to do as much work to get there as you will
--John
An interesting counter-proposal to address this dilemma is proposed by a New Zealand economist named Ronnie Horesh, at http://www.geocities.com/WallStreet/9856/, albeit in a different context. Horesh's idea is a social policy bond issued by a government or somebody else with deep pockets and a social agenda. The bond is redeemable for a large fixed sum when the goal is achieved. Until then, the bond is a good that can be bought and sold like a share of stock. Like a share of stock, its price will rise as the goal moves closer to being accomplished. The purpose of inventing these bonds is to delegate the implementation of social goals to free market forces.
If a mathematician felt confident that he had made an important contribution, but didn't feel confident that he'd complete the proof himself, he could buy bonds cheaply as soon as they were issued, and hold onto them until the proof was complete. When the proof was made public, each bond would become redeemable for a large fixed sum, and the guy who bought early would make big bucks.
WWJD for a Klondike Bar?
Erg. Sorry, wrong answer.
Mathematics is replete with proofs about infinite sets of numbers. My personal favorite (mostly because I actually understand it) is Cantor's hypothesis:
Cantor wanted to prove that even though there's an infinite number of integers, and an infinite number of "real" numbers, there are clearly more real numbers than integers. So he devised this infinitely large list. You take every real number (even ones of infinite length) and write them down on a convenient piece of infinitely large graph paper, one digit per square. Finally, you number each one in the left hand column with a counting number (positive integer).
Now, since you've used graph paper, you can show that there's a real number not on the list. You start at the first real number, take the digit in the first column, and change it to something else. Go to the second number, take the digit in the second column, and change it. And so on, and so forth, blah blah, nth digit from nth number, etc. In the end, you have a new real number which can't be duplicated anywhere on the list, because it's sure to vary from every other number by at least one digit. Furthermore, even if you add the new number to the bottom of the list, you haven't solved the problem: you can make a brand new number which still doesn't appear anywhere on the list just by repeating the process. This kind of proof utilizes what's called a "diagonalization method", since the number you wind up constructing can be viewed by drawing a diagonal line across the list of reals you made up.
Anyway, this is getting way too long. The upshot: there are plenty of examples in mathematics of people making proofs about infinite sets of numbers, and they don't have to sit down with calculators for an infinite length of time to write them. More to the point, since no one actually can enumerate all of the primes (because there are infinitely many of them, see), no computer program is going to be solving this one any time soon.
Well, you can just e-mail that to me and I'll be sure to pass it along to Rob Malda so he can post it elsewhere on the page :) ...just take out the TINLC part, there is no lumber cartel.... :)
My journal has hot
That is - an extremely small probability!!!
:)
Yeah, but close only counts in horseshoes, hand grenades, and Inter-Continental Ballistic Missiles.....
My journal has hot
An odd number is some integer (n*2)+1. There is no integer that will satisify 0 = (n*2)+1. Therefore, zero is not odd (it's even).
A prime number is some integer that is divisble only by 1 and itself. Since a denominator of zero is undefined, zero cannot divided itself. Therefore, zero is not prime.
cpeterso
Andrew Wiles' proof of FLT runs into the hundreds of pages and makes use of vast tracts of number theory, some dating back hundreds of years. It can probably only be understood in full by a handful of top mathematicians.
;-)
Now that FLT has been proven, what about Fermat's comment about his "great proof"? Do modern mathematicians believe he was joking? Or did he have some different, simpler proof? That he would have the same lengthy page proof seems unlikely, but it would definitely fit the description of "too large for the margin of this book"!
cpeterso
try compiling with -lm.
What do you mean "such as this"? The whole point of a proof is that it is deeper than a whole list of numbers with checkmarks beside them. "Yep, 5 is sum of two primes. Yep, 7 is the sum of two primes." is just a demonstration, not a proof.
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"Who Wants To Be A Millionaire" is the American version of a British show. Only... nobody there ever won the big prize (the insurance company that pays the prizes is rather upset).
If we find a counterexample, thereby proving the conjecture false, do we also collect the million?
Time to get a cluster searching, eh?
Well I don't have a copy of the standard, but my fragment compiles without warnings on gcc 2.95.2 using the -pedantic option. Since that's meant to make gcc complain about non-ANSI (i.e. ISO) extensions, does that mean this is a bug in gcc? Are you sure it hasn't been added to the standard since the version you've seen?
Hmm the standard says that ^= and -= are right-associative and have the same precedence. The associativity should force the expression to be evaluated as I claim.
Would you care to provide a reference saying where in the ISO standard I should look to see what you're saying?
perl -e 'fork||print for split//,"hahahaha"'
But = also does not define a sequence point, so by that argument "a=b=c=d=2" is also undefined. What am I missing here?
perl -e 'fork||print for split//,"hahahaha"'
Ok, I lied slightly, it is more normal to say "any *natural* number" rather than "any number". A natural number means 1, 2, 3,
Again, the definitions are tailored specially so that things are neat to state. To a mathematician, "a product" doesn't neccessarily mean two or more numbers. Just "5" by itself gets counted as a product. And the empty product, the product of 0 numbers, is arbitrarily defined as being "1". So the prime factorisation of 5 is "5", and the prime factorisation of 1 is "".
You're probably thinking that the "empty product equals 1" thing is a bit bizzare. It's worth comparing with addition. What do you get if you add zero numbers together? Answer: 0. 0 is the number that makes no difference when you add it to things. Similarly, 1 is the number that makes no difference when you multiply things by it. Hmmm, I'm not sure if that makes it any clearer, but hey.
perl -e 'fork||print for split//,"hahahaha"'
Factorising isn't exactly very "useful". It's more that it's kind of the underlying thing that causes the numbers to work like they do. Without the Fundamental Theorem, mathematicians couldn't have any sort of a grip on what the hell multiplication "is". With the Fundamental Theorem, we know that multiplication has this nice, orderly property which allows us to understand it well enough to prove more difficult stuff.
So you're right, to an engineer, or a scientist, factorization isn't very useful, because these people just "use" arithmetic, as a sort of contraption that works. The Fundamental Theorem is only Fundamental to a mathematician, who is interested in studying the contraption itself.
If a and b are two non-zero integers, it replaces a with the highest common factor of a and b (clobbering b in the process).
perl -e 'fork||print for split//,"hahahaha"'
Who says mathematicians can't write poetry? Hmmm.
perl -e 'fork||print for split//,"hahahaha"'
"Words have their meanings to help you understand", as Lauren Savoy would have it. The word "prime" is defined in the way which makes the word most useful. The central fact about primes is that "any number can be expressed uniquely as a product of prime factors" (ignoring the order in which you write the factors). This is regarded as so fundamental that it's called "The Fundamental Theorem of Arithmetic".
If we were to accept 1 as a prime, there'd be lots of ways to prime-factor a number, e.g. 9 = 3*3 = 3*3*1 = 3*3*1*1*1*1. So to keep the Fundamental Theorem true, we'd have to say "except 1" in it. There's also lots of other theorems where you'd need to say "except 1" for similar reasons, so it works out as more convenient to exclude 1 when defining the word "prime".
For this reason also, we exclude negative numbers from being prime. Otherwise, 9 = 3*3 = -3*-3 etc.
perl -e 'fork||print for split//,"hahahaha"'
This is true; IIRC it was proved by Ramanujan.
Another useful fact: (number of primes less than N) tends to N / log(N) as N tends to infinity. This is called the prime number theorem.
perl -e 'fork||print for split//,"hahahaha"'
they count as integers :)
but I dont think so
How we know is more important than what we know.
you need a generalised proof.. something like induction.. betchya wished you'd paid attention in math class now!
How we know is more important than what we know.
what I did first was to figure out the number of primes that sum to the first 20 or so evens.. obviously it's been done before (hey, I was hoping for a fibonachi sequence or something groovy like that).. anyways
.att.com/~njas/sequences/Sindx_G.html#Goldbach
http://www.research
How we know is more important than what we know.
The k p (calc-prime-test) command checks if the integer on the top of the stack is prime. For integers less than eight million, the answer is always exact and reasonably fast. For larger integers, a probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P). The number is first checked against small prime factors (up to 13). Then, any number of iterations of the algorithm are performed. Each step either discovers that the number is non-prime, or substantially increases the certainty that the number is prime. After a few steps, the chance that a number was mistakenly described as prime will be less than one percent. (Indeed, this is a worst-case estimate of the probability; in practice even a single iteration is quite reliable.) After the k p command, the number will be reported as definitely prime or non-prime if possible, or otherwise "probably" prime with a certain probability of error.
How we know is more important than what we know.
Faber has stipulated that the proof must be submitted to a respectable mathematical journal within two years of the book's publication next week, and published within four years. A panel of world-renowned mathematicians will be appointed to decide whether the proof is valid (Faber is refusing to disclose the panel's names, for fear that they will be flooded with letters from amateurs).
I have a serious question here, and I hope that someone might be able to answer it.
From the tone of the article, it sounds like they are expecting somebody in the mathematical community to submit it, if anybody does. How would say, me, your average slashdot reader, go about claiming such a prize? I have no idea how one would go about publishing something in a mathematical journal, and with such a prize at stake how would one prevent one's proof from being stolen by someone else (like a person you find to "help" you get it published)?
I'm not a journalist, but I play one on slashdot
It does make sense to have lots of people offering lots of little prizes all over the place.
As I've pointed out in The Bowery Prize for Amateur Rocketry:
The Internet provides a unique opportunity to promote space technology via prize awards. Individuals can post a public pledge of money as a prize award for any technical objective they see fit, to be disbursed at their sole discretion. With enough diversity of people and technical objectives, there would be a "fuzzy" gradient of incentive created for ever higher performance amateur rockets, not dependent on the credibility of any one organization's political structure for "fairness" or good technical judgement. A periodic posting of all such prize award offers in this, and related, newsgroups, would serve as an ongoing challenge to technical excellence and as an inspiration for young people.
This sort of approach to what should be called "Contest Philanthropy" will address the concern that people will hold off publishing important but obscure work.
Here's how:
If a bunch of mathematicians believe a particular popular conjecture is critically dependent on some obscure, but hard, math getting done, then all they need to do is post a prize award that promises they will give half their winnings for the popular conjecture to the mathematician who does the obscure work. If enough credible mathematicians target a particular obscure piece of work, then the philanthropists can start providing direct prize awards for that support work.
Seastead this.
Unfortunately, that does nothing to make finding prime numbers any easier. IE, knowing the approximate shape of the graph (and the graph isn't perfect) still doesn't give you any information for calculating the next prime number. You have to pick a number (ending in 1,3,7, or 9), brute force it to test its primitude (I just made up a word!) by dividing it by all previous primes = the square root of the number you're testing.
-CausticPuppy "Of all the people I know, you're certainly one of them." -Somebody I don't know
For those banging their heads against the monitor. He multiplies by zero at one point. Then, all bets are off.
But your assumption is wrong. How about:
1) Assume 1 == 1
2) Either 1 == 1 or N = 4 for all N
3) 4 is the sum of two primes
4) by 2 and 3, for all N, N is the sum of two primes
5) Since all numbers are the sum of two primes, all odd numbers are the sum of two primes (QED)
Finding that number is left as an exercise for the reader. Heh.
Best regards,
SEAL
An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.
Thus, negative numbers are not prime.
Best regards,
SEAL
Maybe.. maybe not. Think Fermat's last theorem. This would be the obvious approach for that too except that a proof was found differently(elliptic curves and all that cool stuff). Primes are a little more trickier to deal with than regular integers, though. Besides, according to the article someone has already proven that all odd integers starting from some sufficiently large number are the sum of three primes. Now just reduce the number to 5, add even numbers and collect million bucks.. think I'll try it tonight on my lunch break.. not.
Slashdotter's should consider reading "Brill's Content" magazine for its recent story on the media's incompetent handling of the "Gore invented Love Canal" story. Then they should ask themselves if a guy as smart as Gore, who actually does have well-reported techno-wonk credentials, would really have claimed to have "invented the Internet."
Gore was one of the leading proponents of the "information superhighway." Now, if he wants to talk more about it, he runs the risk of a bunch of know-it-nothings laughing at him and hurling once-funny cheap shots.
This "Gore says he invented the Internet (har-har)" stuff is a silly mental virus propagated by people with relatively weak intellectual immune systems, IMHO. That's not to say they are stupid, just susceptible. Gore fouled up his line, true. But anyone interested in Slashdot should still be for him.
Ah, but you can't say that a formal proof is the correct method unless you have proved that a proof exists.
In which case, you have proved the conjecture and should be collecting your prize rather than posting to slashdot.....
If on the other hand the above proof does not exist, then there must exist a counterexample. Therefore searching by brute force may be a perfectly valid approach.
Hmmmm.... you can't actually *prove* Goldbach's conjecture by direct computation because the number of combinations you would have to try is infinite. It's the same reason why you couldn't prove Fermat's Last Theorem this way.
:-)
You could of course *disprove* the conjecture by finding a counter-example, i.e. by finding a positive even number X>2 such that X is not the sum of two primes.
It would certainly be fun if you were to find such a beastie, but I think you might be in for a long search
you wrote The word "prime" is defined in the way which makes the word most useful. The central fact about primes is that "any number can be expressed uniquely as a product of prime factors" (ignoring the order in which you write the factors). This is regarded as so fundamental that it's called "The Fundamental Theorem of Arithmetic".
This fundamental theorem seems to me to be wrong in principle, it requires 1 not to be a prime, where does 0 fit in (or is it not a number). What about the primes themselves if 1 is not a prime, or have you just mis-stated the theorem?
What are the prime factors of 5?
no sig.
The trick is to use an abstract and generalised proof, for example:
by contradiction: you say that the conjecture is untrue, then extrapolate some fact from this which is patently untrue. Hence the initial conjecture must be true.
by induction: you show that for some instance n the conjecture is true. You then show that for the general step n+1 the conjecture must also be true through the use of some logical argument, rather than just testing it. Hence the conjecture must be true for all instances.
abstraction: find some analogous system and proove it for that. For example, when doing analysis of AC curcuits the common technique is to consider the current and voltage as phasors (ie. V*e^i*theta)and manipulate the phasors which is much simpler. When the manipulation is complete you can simply take the real/imaginary part of the result, and you have your answer. This makes use of the identity that e^i*theta = cos theta +i*sin theta. (I shall leave the proof as an excercise for the student [grin])
"The new wave is not value-added; it's garbage-subtracted" - Esther Dyson, Dec 1994
1 isn't a prime number, as this would cause the natural numbers not to be a unique factorization domain.
at least in the UK you seem to value education and things such as this. Here in the states we prefer to have people win a million dollars by either picking 7 random numbers and watching ping pong balls or answering 10 questions or so from Regis.
Goldbach's conjecture is very probably true; mathematicians and number theorists don't need to be reminded of that and likewise don't need to be bothered by "amateurs" (really, lay-people without formal education in number theory) who suggest that the conjecture is false. In all matters of fact, while I am sure that Faber and his erudite associates would be happy to be presented with a valid counter-example, said mathematicians probably aren't losing any sleep over the possibility. What Tony Faber is looking for and what's worth $1m to him is not the labors of the unitiated towards a hopeless cause but is instead the establishment of the conjecture's logical truth by means of established principles of number theory -- he wants to know how this elegantly simple and ostensibly true statement may be derived from postulates and previous proofs, not, regrettably, whether or not it is true.
There has been some grumbling over the apparent fact that Faber is trying to keep the proof of Goldbach's conjecture within the mathematical community. If the conjecture is proved, it will be through the collaborative efforts of the professional mathematic community -- I simply can't see someone without extensive background in number theory really having a chance of winning the contest. Remember that the proof of Fermat's theorem was by no stretch of the imagination simple and elegant. If there were an elegant proof to Goldbach's conjecture, it would almost certainly have already been realized by mathematicians who are just as intelligent and, if anything, better-versed in matters of number theory than folks like you and me, and if there is a proof, I suggest that its development will not be a feat accomplishable by anyone who doesn't make their living in mathematics.
And before anyone tries to make this proof a black&white, true/false issue (too late?), let us remember the work of mathematician Kurt Gödel:
Do I underrate the ingenuity of the human mind? I hope so; however, I would like to think I am merely being realistic. It seems to me that blind beginner's luck has been drastically overrated in response to the proposed contest.
To those of you interested by the prospect of testing every reasonably-large even number to find a counter-example and thus disprove the conjecture, and especially to those of you who have suggested that the resources of distributed.net be used in so doing, please refer back to sela, 3/19. I'm afraid you'll be wasting your time.
_subterfugitive
Do negative numbers count as primes?
I see even classic Slashdot is now pretty much unusable on dial up anymore.
Honest,
Al Gore told me he's got the proof to this theory - he did it while he was inventing the Internet, and Open Source and all that.
1) Assume 1 == 0.
2) Then clearly this follows Either 1 == 0 OR N = 4 for all N
3) Obviously 4 is the sum of two primes (2+2)
4) Combining steps 2 and 3 we get For all N, N is the sum of two primes
5) Step 4 states that all numbers are the sum of two primes, therefore all odd numbers are the sum of two primes.
QED
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The first step any Goldbach-conjecture-prover-wannabee should take is to read the book ``Additive Number Theory: the Classical Bases'' by Melvyn B. Nathanson, in Springer's Graduate Texts in Mathematics (number 164). It contains a description of the sieve methods and the Hardy-Littlewood-Ramanujan circle methods used to prove that sort of results in additive number theory. It gives a proof of the closes results known to Goldbach's conjecture: Vinogradov's theorem that every sufficiently large odd integer is the sum of three primes, and Chen's theorem that every even integer is the sum of a prime and a number which is at most product of two primes.
If you think it must be interesting reading, you're dead wrong. I have read many boring proofs in my life, but none of them came even close to the total lack of interest, the absolute and utter dullness of the analytic estimations of additive number theory, as found in the book in question. Even the (much easier) proof of Waring's problem (every integer is the sum of a bounded number of k-th powers, where the bound depends only on k) is unconceivably dull, and it is nothing compared to the more subtle results in the second part of the book.
Now why couldn't they offer a million dollars for proving the Riemann hypothesis. At least that's a worthwile result (nobody cares in the least about Goldbach's conjecture, but the Riemann hypothesis has many fascinating consequences).
// usage: goldbach
// 2 is prime
// Lists all pairs of primes that sum to
// Copyright Chaz Randles 2000
// Licence: GPL
#include
#include
using std::cout;
using std::endl;
using std::vector;
vector * get_primes(int maxPrime) {
vector* result = new vector();
vector::iterator it;
bool isPrime;
result->push_back(2);
for (int candidate = 3; candidate begin(); it!=result->end(); ++it) {
if (!(candidate % *it)) {
isPrime=false;
}
}
if (isPrime) result->push_back(candidate);
}
return result;
}
bool find(vector* vec, int data) {
vector::iterator it;
bool found=false;
for (it=vec->begin(); it!=vec->end(); ++it) {
if (*it==data) {
found=true;
break;
}
}
return found;
}
int main(int argc, char** argv) {
if (argc!=2) {
cout " * primes=get_primes(num);
vector::iterator it;
for (it=primes->begin(); it!=primes->end(); ++it) {
if(find(primes, num - (*it))) {
cout num '\t'*it'\t'num - (*it)endl;
}
}
delete primes;
return 0;
}
// Apologies for poor quality code - done in 10 mins
No, you can do tricks like "proof by contradiction". That works as follows:
perl -e 'fork||print for split//,"hahahaha"'
That means that "there is some way to get to any even number by adding together less than 300,000 primes". It doesn't mean "no number can be obtained by adding more than 300,000 primes". The second statement is clearly false. e.g. if
then a is the sum of 300,001 primes.
perl -e 'fork||print for split//,"hahahaha"'
Lots of work has been done that could be useful to the GC, just the same was as done for FLT. The winnings will go not to the person who does the most work towards it, but whoever supplies the piece of the puzzle that happens to be last.
Mathematics is a great cooperative venture. It wouldn't be easy to identify one mathematician who did the largest share of the work even if they tried to do it that way, and if they did it would probably be Gauss or Poincare or some other dead heavyweight genius.
This might encourage work on the GC, but it also might discourage publication of such work, because the mathematicians haven't quite finished the proof.
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Xenu loves you!
Unfortunately, the slashdot comment submission box is too small to post the whole proof. I shall leave the proof to the reader.
"Pinky, you've left the lens cap of your mind on again." - P&TB
"I can see my house from here!" - ST:
Yes, it is possible that the Goldbach conjecture may be proved or disproved by a lay person but it's far more likely to be solved by someone with a sound grasp of lots of number theory who is already studying the properties of numbers for a living.
BTW I have a truly marvellous proof of the Goldbach conjecture but there is no margin on the submit form to write it in :)
--- Hot Shot City is particularly good.
Disclaimer: long boring math stuff with a conclusion that may be interesting for some at the end ...
...
...
An interesting question is: assuming prime numbers are "randomly" distributed, what is the probability that an even number won't be the sum of two odd numbers?
If the probability is rather high, yet we did not find any such even number, it can be an indication there may be some "meat" in goldbach theorem. On the other hand, if such probability is exremely low, than the verification of goldbach's theorem up to 4e+15 means nothing
Well, according to Gauss' estimate for prime number density around sufficiently large n, the density is 1/log(n). To find pi(n), which is the number of prime numbers up to n, we need to integrate this, but pi(n)=n/log(n), so we can make life simpler and over-estimate the probability by taking the smaller number.
Now, given the number of prime numbers up to n, pi(n), if we choose x to be a prime number smaller than n, than the probability its pair: (n-x) is a prime number is: 2*pi(n)/n. (I multiplied by 2 since itcannot be possibly an even number, so no need to consider them).
The probability that it is not a prime number is therefore (1-2*pi(n)/n).
Since there are pi(n) different prime numbers to choose from, the probability that an even number is not the sum of two prime numbers is:
(1-2*pi(n)/n)^pi(n).
So what is the probability for numbers around 1e+15? If we use the estimate for pi(n), we get:
pi(n)=2e+13
(1-2*pi(n)/n) = 0.959
Taking the last number by the power of pi(n), the result is that the probability is around:
10^-(4e11)
That is - an extremely small probability!!!
Conclusion:
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If Goldbach's theorem is false, the probability that an even number won't be the sum of two prime numbers is so small that we may need to check around 10^(10^11) numbers to find a counter example. This number is so big no computer could possibly handle it now or any time soon
In other words: Don't bother to find a counter-example.
Sela
It's stunning how many people want to approach this as a programming problem; i.e. a fast way of checking lots of numbers. We're talking about a *proof*. Guessing and checking is missing the point.