Major Advance Towards a Proof of the Twin Prime Conjecture 248
ananyo writes "Researchers hoping to get '2' as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million. That goal is the proof to a conjecture concerning prime numbers. Primes abound among smaller numbers, but they become less and less frequent as one goes towards larger numbers. But exceptions exist: the 'twin primes,' which are pairs of prime numbers that differ in value by 2. The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart. He presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever."
Preprints not avaiable, but it seems legitimate (Score:5, Informative)
The paper seems to have been accepted by Annals of Mathematics, which is basically the number one mathematics journal.
Also, according to New Scientist, Henryk Iwaniec (a well-known analytic number theorist) has reviewed the paper and didn't find an error. This may or may not overlap with the review at Annals, though.
Re:Open set it is! (Score:5, Informative)
Imagine that you did find all the prime numbers, every single one.
Then, take them, and multiply them all together.
Add 1.
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Re:Open set it is! (Score:3, Informative)
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
Re:Open set it is! (Score:5, Informative)
GP has already used all the supposed finite number of prime numbers in constructing his contradictory bigger prime.
The proof constructs a number that is not divisible by any of the prime numbers in the set of all prime numbers. Therefore it proofs there are an infinite number of prime numbers. The conclusion the constructed number must be prime is wrong.
Nyh
Re:Open set it is! (Score:5, Informative)
The GP's correction is right.
The GGP said that his number was prime. It might be, but it might not. But if it's composite then it cannot be divisible by any of the primes in his initial set so there must be a prime not in his set.
For example, if we assume 13 is the last prime then multiply them all together and add 1 we get 30031. But 30031 is not prime - it's divisible by 59 (which is a prime not in our set)
Tim.
Re:Primes closer together? (Score:5, Informative)
Not quite.
This means that for every prime p such that p+q (where q is less than 70 million) is also prime, there exists another prime r bigger than p such that r+s (where s is also less than 70 million) is also prime. Note that there is no limit to the distance between p and r.
Re:Open set it is! (Score:2, Informative)
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
No, the assumption was that you have all prime numbers. You're not allowed to violate assumptions within a formal proof, you're only allowed to point out self-contradictions.
Conclusion wrong (Score:4, Informative)
"the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever"
Actually, I disagree with the unfortunate writing of the sentence. The gaps between consecutive prime numbers are variable, and on average they DO tend to keep growing forever. This is a widely known result, the density of prime numbers decreases as the numbers grow. However, since the gap between consecutive primes is variable and it does not follow a regular function (otherwise, it would be very easy to calculate prime numbers), even with a very low density of prime numbers we can find a pair of consecutive prime numbers with a gap of only 2.
The problem under study is not wether the gap between consecutive primes keeps growing forever (which is true only on average, considering a long secuence of integers), but wether there are infinite such pairs of primes with gap 2. The new result found says that there exist infinite pairs of primes with gap 70M or less. However, this does not imply at all that no consecutive pairs of primes with gap > 70M exist (which, in fact, they do).
Re:Gaps between numbers... (Score:5, Informative)
Joking aside, submitter is not a mathematician. This doesn't prove anything about the gap between arbitrary consecutive primes. That gap does indeed grow forever, by the known distribution of primes, but by "chance" one would expect a few pairs to lie close together. The proof is that this "chance" event still occurs as N tends to infinity. The same result would hold for random numbers whose distribution gets more sparse with increasing N so it just says that the primes are not "less random" than these (in a very informal sense).
Re:Open set it is! (Score:1, Informative)
His proof is fine, you're just not familiar enough with how proofs by contradiction work. It's true that 30031 is composite. Within the formal context of his proof by contradiction, it is simultaneously true that 30031 is prime. It is prime because it is not divisible by any other prime. It is composite because you found a non-trivial factorization. Both statements are true. That's why it's a proof by contradiction - there is a contradiction. Your wording is correct too, but it is not necessary to add the case that you did where X+1 is not prime, though that extra bit may be helpful for less mathematically experienced readers.
The Question (Score:5, Informative)
Researchers hoping to get '2' as the answer
In case anyone's as confused as I was, I think I've finally figured out The Question, which is:
What is the smallest gap between consecutive primes which occurs infinitely many times?
Or something like that. Everyone thinks it's probably 2.
Re:Open set it is! (Score:4, Informative)
Euclid's Theorem in actuality [clarku.edu] does refer to the case where X+1 is not prime. It's essential to the proof.
It goes something like this:
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Take a finite list of prime numbers, A, B, C etc. (The assumption that they are "all the primes" is unnecessary.)
Find the smallest common multiple of them, X.
Add 1 to that.
The new number, X+1, is either prime or composite.
If it's prime, then that's it. We've generated a new prime not on the list.
If it's composite, then it is divisible by some prime, G.
Could G be one the primes (A, B, C. etc.) already on the list?
But remember, X is divisible by A, B, C etc. So if G is one of those primes, then that means that both X and X+1 are divisible by prime number G, which is impossible.
Therefore G would have to be a new prime, not on the list.
Now we have a larger list, A, B, C, G, etc. and can repeat the process.
We can always generate a new prime not on the list, and therefore the list of primes is without bound.
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