# A Math Question **URL:** \ **Category:** Factual Questions\ **Created:** [March 28, 2018, 4:55am UTC](https://boards.straightdope.com/t/a-math-question/811323 "2018-03-28T04:55:44Z")\ **Posts on this page:** 17\ **Page:** 1
**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\ **Post date:** [March 28, 2018, 4:55am UTC](https://boards.straightdope.com/t/a-math-question/811323/1 "2018-03-28T04:55:44Z")
A Math Question I am far from certain that I understand the point of Yitang Zhang’s recent (2017) prime number discovery. Is this correct? 1. 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). 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. 3. Zhang proved that there is some gap between two primes that is shorter than seventy million and is infinite. Is this what Zhang proved? ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [March 28, 2018, 5:30am UTC](https://boards.straightdope.com/t/a-math-question/811323/2 "2018-03-28T05:30:22Z")
The underlying problem that he was working on is the Twin Prime Conjecture. See here: > **[Twin prime](https://en.wikipedia.org/wiki/Twin_prime#History)** > > A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. > Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as th... (I assume you mean his 2013 publication.) ---
**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\ **Post date:** [March 28, 2018, 8:09am UTC](https://boards.straightdope.com/t/a-math-question/811323/3 "2018-03-28T08:09:01Z")
Rely t Reimenn No. Zfan’s new 2017 discovery. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 28, 2018, 12:55pm UTC](https://boards.straightdope.com/t/a-math-question/811323/4 "2018-03-28T12:55:47Z")
> [@Steven\_Estes](#): > > No. Zfan’s new 2017 discovery. I haven’t heard of a new 2017 discovery and can’t find a reference to one online. Got a link? ---
**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\ **Post date:** [March 28, 2018, 1:52pm UTC](https://boards.straightdope.com/t/a-math-question/811323/5 "2018-03-28T13:52:43Z")
Try these. There are others, but they are overwhelmed by sites discussing 2013 discovery > **[A Huge Discovery About Prime Numbers—and What It Means for the Future of Math](https://slate.com/technology/2013/05/yitang-zhang-twin-primes-conjecture-a-huge-discovery-about-prime-numbers-and-what-it-means-for-the-future-of-math.html)** > > Last week, Yitang “Tom” Zhang, a popular math professor at the University of New Hampshire, stunned the world of pure mathematics when he announced... # “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.” ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 28, 2018, 2:07pm UTC](https://boards.straightdope.com/t/a-math-question/811323/6 "2018-03-28T14:07:47Z")
> [@Steven\_Estes](#): > > Try these. There are others, but they are overwhelmed by sites discussing 2013 discovery > > [Yitang Zhang, twin primes conjecture: A huge discovery about prime numbers—and what it means for the future of math.](http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html) 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. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [March 28, 2018, 2:38pm UTC](https://boards.straightdope.com/t/a-math-question/811323/7 "2018-03-28T14:38:39Z")
> [@Thudlow\_Boink](#): > > 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. 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! ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [March 28, 2018, 2:41pm UTC](https://boards.straightdope.com/t/a-math-question/811323/8 "2018-03-28T14:41:31Z")
> [@Thudlow\_Boink](#): > > for large n, n!+2 must be divisible by 2, n!+3 must be divisible by 3, etc. I get why you wrote “for large n”, but for anyone who didn’t notice: for all n\>1 it is true that all the numbers from n!+2 to n!+n are non-prime. It doesn’t _have_ to be a large n. Apologies if this is some sort of math version of the astrophysicist table of elements. The one that goes Hydrogen, Helium, Metals. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 28, 2018, 2:42pm UTC](https://boards.straightdope.com/t/a-math-question/811323/9 "2018-03-28T14:42:50Z")
> [@Steven\_Estes](#): > > A Math Question > > I am far from certain that I understand the point of Yitang > Zhang’s recent (2017) prime number discovery. > > Is this correct? > > 1. 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). > 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. > 3. Zhang proved that there is some gap between two primes that is shorter than seventy million and is infinite. > > Is this what Zhang proved? 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. ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [March 28, 2018, 4:06pm UTC](https://boards.straightdope.com/t/a-math-question/811323/10 "2018-03-28T16:06:28Z")
> [@Exapno\_Mapcase](#): > > 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. After the paper came out there was a fury of activity and based on Zhang’s work people go the number down to 246. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [March 28, 2018, 9:24pm UTC](https://boards.straightdope.com/t/a-math-question/811323/11 "2018-03-28T21:24:36Z")
> [@gazpacho](#): > > After the paper came out there was a fury of activity and based on Zhang’s work people go the number down to 246. 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. ---
**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\ **Post date:** [March 29, 2018, 4:47am UTC](https://boards.straightdope.com/t/a-math-question/811323/12 "2018-03-29T04:47:14Z")
“Which is to say that there are infinitely many pairs of primes that differ by at most 246.” Does this also mean that there are infinitely many pairs of primes that differ by at most 247, 248, 249… 1,600,000…? ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [March 29, 2018, 5:05am UTC](https://boards.straightdope.com/t/a-math-question/811323/13 "2018-03-29T05:05:16Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 29, 2018, 1:48pm UTC](https://boards.straightdope.com/t/a-math-question/811323/14 "2018-03-29T13:48:08Z")
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. ---
**Author:** ![Ponderoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ponderoid/32/19221_2.png) [@Ponderoid](https://boards.straightdope.com/u/Ponderoid)\ **Post date:** [March 29, 2018, 3:59pm UTC](https://boards.straightdope.com/t/a-math-question/811323/15 "2018-03-29T15:59:49Z")
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? ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [March 29, 2018, 4:03pm UTC](https://boards.straightdope.com/t/a-math-question/811323/16 "2018-03-29T16:03:50Z")
(p, p+k) does NOT occur more than once if k is odd. (p, p+k) is conjectured to always occur infinitely often when k is even. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [March 29, 2018, 5:18pm UTC](https://boards.straightdope.com/t/a-math-question/811323/17 "2018-03-29T17:18:33Z")
> [@Ponderoid](#): > > 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? As mentioned above, it’s easily provable for odd values of k. It hasn’t been proven either way for infinitely many even values of k. It would be a nifty result though and would disprove Polignac’s conjecture.