I am far from certain that I understand the point of Yitang
Zhang’s recent (2017) prime number discovery.
Is this correct?
If you go out far enough on the number line, you’ll get to a gap of, say, a billion numbers between two prime numbers, (say, a billion even numbers that don’t include 2).
There are also prime numbers that are, say, just two apart (3,5), and many others with short gaps, but it is not known if any of these with shorter than seventy-million gaps between two prime numbers are infinite.
Zhang proved that there is some gap between two primes that is shorter than seventy million and is infinite.
Try these. There are others, but they are overwhelmed by sites discussing 2013 discovery
“Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. In other words, that the gap between one prime and the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture.”
Not sure of web address
“Zhang’s discovery demonstrated that the number of prime pairs that are less than 70 million units apart is infinite.”
“…Zhang has managed to prove a weaker** form of the Twin Prime Conjecture, that there are an infinite number of prime numbers no more than some fixed number, between 2 and 70 million, apart. “What!?!? 70 million!?!?” I hear you cry. “You’re aiming for an answer of 2 and you’re at 70 million, this isn’t news.”
That link is to a 2013 article. So it may be that that’s what you’re thinking of, and that there’s nothing new from 2017.
If so, then: For hundreds of years it has been a famous open question whether or not there are infinitely many pairs of prime numbers that differ by two (“twin primes” like 3 and 5, or 29 and 31). What Yitang Zhang did in 2013 was prove that there are infinitely many pairs of primes that differ by, not two, but a different (and much larger) specific number—which is a big deal because nobody else had managed to prove anything like that before.
By the way, it’s fairly easy to show that there are arbitrarily large gaps between primes (i.e. arbitrarily long runs of composite numbers within the sequence of natural numbers), based on the fact that, for large n, n!+2 must be divisible by 2, n!+3 must be divisible by 3, etc.
My impression is that neither Zhang nor anyone else has demonstrated such a specific number; merely that that such specific numbers exist, and have proved upper bounds for the smallest such numbers. It is currently known that at least one number k in {2, 4, 6, 8, …, 246} produces an infinite number of prime pairs (p, p+k). If the Generalized Elliott–Halberstam Conjecture is true than there is a satisfactory k in {2, 4, 6}. If Polignac’s Conjecture is true than every even k is satisfactory.
But AFAICT not a single specific satisfactory k is known for sure.
To understand how stupendous Zhang’s result is, note that small primes — say any less than a googol-plex or Graham’s number — are irrelevant. There are an infinity of prime (p, p+k) pairs even for big numbers, where the mean distance between primes is greater Graham’s number!
Your wording confuses me, but Zhang proved there are an infinite number of gaps *at most *70,000,000. The idea is to keep lowering the boundary - 70,000, 7000, 700 etc, - until the real prize - whether an infinite number of twin primes exist - is found or disproven.
Which is to say that there are infinitely many pairs of primes that differ by at most 246. I knew it had been reduced but I don’t know how low it had gotten. And, believe it or not it was considered an amazing result.
Trivially, yes. If they differ by 246, they are already differ by at most 247 and most. That doesn’t mean N=246, though. It means there exists (though we don’t know which one in particular) some N less than or equal to 246 such that there are infinitely many primes with a gap of N.
But if you mean by exactly 248, 250, 252, 1000, or any particular N (it does have to be an even number by the way), then no, the general result (aka Polignac’s conjecture) hasn’t been proven or disproven for general N.
And of course, the suspicion is that k=2 (that being the Twin Prime Conjecture). We still haven’t proven it, but we seem to be a lot closer to proving it than we used to be.
Has anyone ever proved that some possible prime gap k where k>1 does not appear an infinite number of times? For any particular k, or just that some k with that property exists?