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."
Stories like this... (Score:2, Interesting)
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
Primes closer together? (Score:0, Interesting)
Also means that there must be at least one prime in every sequence of 70 million integers.
Means I can put an upper bound on my prime search script....If it searches 70,000,001 consecutive integers and claims to have found no primes, I'll know the bugged little script is lying.
That's a helpful debugging heuristic. Thank you, Pure Math.
Re:Open set it is! (Score:4, Interesting)
Or more elegantly in haiku form:
Top prime's divisors'
product (plus one)'s factors are?
QED, bitches!
-- http://xkcd.com/622/ [xkcd.com]
Re:Open set it is! (Score:5, Interesting)
From a purely mathematical point of view you are incorrect.
The proof isn't that there's less than 70million units between each prime (like there's a lot of primes with a gap of two units eg 29 and 31, 41 and 43 etc). the proof is that there's in infinite number of prime pairs with a maximum of 70 million units between them.
You can still find gaps significantly larger. Those gaps are present between numbers that are NOT prime pairs.
eg: 29 30 31 32 33 34 35 36 37 39 40 41 42 43 44
Here there is a prime pair with a 2 unit gap between them (41 and 43), however the number 37 has a larger gap on either side, because it is not a part of a "prime pair". In your thinking you are excluding the primes that are NOT paried, and the gaps between where one pair ends and another begins. Each of which, according to the proof still has the ability to exceed 70 million units.
Disclaimer: I did not fully read the proof posted in annals of mathematics, but I'm pretty certain that this is the gist of it