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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. "
If you disprove the theorem then you pay them $1 million.
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
Having said all that, the other poster was right; if this were easy someone would have done it already. At the very least you will need enough mathematical background to write with the correct terminology and symbols. Really you shouldn't seriously attempt this sort of research without first becoming familliar with what other people have tried and why it didn't work. Please don't annoy the journal editors with bogus submissions (not that I think you were planning to, but it bears repeating nonetheless).
-rpl
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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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 :)
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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:
----------
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.