Number Theory Awesomeness

"On May 13, an obscure mathematician garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers. Yitang Zhang showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million. Then Terence Tao, a winner of the Fields Medal, mathematics' highest honor, created a 'Polymath project,' an open, online collaboration to improve the bound that attracted dozens of participants. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600."

I always thought number theory and related proofs were pretty neat. But I can't even complrehend how you prove there are an infinite number of prime pairs spereated by no more than 70 million, and then interatively advance the gap down to 60 million, 4680 and 600.  I'm too stupid to understand the proofs (and too smart to try), but the gap sizes seem extraordinarily arbitrary for number theory proofs.  It's like saying "I can't prove that there are finite cows in the world, but i CAN prove there are at least 294".  It's not really like that at all, but that's the best I could come up with.


Bonus number theory proof joke:

Several scientists were asked to prove that all odd integers higher than 2 are prime.

Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime,...
Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, therefore by induction......

http://science.slashdot.org/story/13/11/20/1256229/mathematicians-team-up-to-close-the-prime-gap