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Major Advance Towards a Proof of the Twin Prime Conjecture
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."
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.
No siree. Ain't non prime numbers at all here in North Carolina since we done banned them. Ain't no angels felled out of the sky, ain't no computers breakin', and my cousin's kisses never tasted sweeter. Prime numbers are a godless socialist conspiracy against Jedus and mah wallet.
To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.
There is a simple, ancient, proof that there are infinite prime numbers.
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.
"First they came for the slanderers and i said nothing."
This is probably the worst written summary that I have ever read on Slashdot.
You must be new here.
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
Don't feel bad. Maybe you've made coffee for, served fries to, or unclogged the toilet of one of these great people? Every little bit helps!
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.
systemd is Roko's Basilisk.
It's shoulders all the way down.
Um, one question that a person could ask is: If this proof is found, how does it change the world? How would being able to use the proof influence something in the real world? I'm not saying it can't or won't, only that simply picking a brainy subject does not mean that doing things in it aren't basically intellectual masturbation.
The change to our world is this: we now know something that we didn't know before. Now we can teach this new knowledge to others (and by others I mean people smarter than me) who can find new places and ways to apply this new knowledge. They might never do anything interesting with it, or it might cause an avalanche of new findings, we don't know. But we, as a species, fundamentally know more today than we did yesterday.
As an example, the ancient greeks studied prime numbers. Was there any immediate use of primes at the time? Did it allow them to improve harvest? Defeat the Roman army? Nope, they just studied them. At the time there is no way that they could have conceived their application for encryption. Yet today, all commerce on the web uses the mathematics of primes.
It is not important to have an immediate use for knowledge.