Big news from the math world:
One of the oldest problems in number theory is the twin primes problem: Are there or are there not infinitely many ways to write the number 2 as a difference of two primes? You can, for example, write 2 = 5 -3, or 2 = 7 – 5, or 2 = 13 – 11. Does or does not this list go on forever? There are very strong reasons to believe the answer is yes, but many a great mathematician has tried and failed to find a proof.
Here’s a related problem: Are there or are there not infinitely many ways to write the number 4 as a difference of two primes? What about the number 6? Or 8? Or any even number you care to think about? It seems likely that the answer is yes in every case, though no proof is known in any case. But….
Just a few days ago, Professor Yi Tang Zhang at the University of New Hampshire announced a major breakthrough. He can’t prove that the answer to any of these particular questions is yes, but he can prove that the answer to at least one of them is yes. Professor Zhang has proved that there is at least one even number — – in fact, at least one even number less than 70 million — that can be written as a difference of primes in infinitely many ways. There seems to be good reason to hope that a refinement of his methods will allow a substantial reduction in that bound of 70 million. Of course if you could reduce the bound all the way to, say, 3, then you’d have solved the twin primes problem.
I thought about trying to explain a little about Zhang’s methods (or about my limited understanding of them), but I cannot hope to do better than this blog post by Emily Riehl.