Gaps between Primes - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as a source for the 70 million bounded prime gap result when answering questions about twin, sexy and other prime pairs.
Recommended by 0 people across 0 different months, 2013 to 2018.
What it is
A short Numberphile explainer from 2013 on Yitang Zhang's breakthrough showing there are infinitely many pairs of primes separated by a bounded gap (under 70 million), a step toward the twin prime conjecture. Brady Haran's channel presents it in plain terms, with a longer discussion linked from physicists Ed Copeland and Tony Padilla.
Cited in 1 Wikipedia article, including Yitang Zhang.
What people said
Cites AncientSwordRage · 2013-05 · Mathematics Stack Exchange · on “Are there practical applications to the new prime pair proof?”
I've recently heard [this video] that its been proven that the set of prime pairs that are separated by no more than 70,000,000 is infinite. Does this have any impact on cryptography or another practical application?
Cites Ovi · 2013-05 · Mathematics Stack Exchange · on “Question about new proof in number theory?”
Very recently, Yitang "Tom" Zhang from the University of New Hampshire made a huge step toward proving the twin prime conjecture. He proved that there are infinitely many pairs of prime separated by a constant distance, and that distance is less that $7 \cdot 10^7$. For example, in the special case of the twin prime conjecture, pairs of primes are separated by 2, such as 3 and 5, 5 and 7, 11 and 13, etc. Now people are saying that they might be able to use his methods to prove that the difference is lower than 16. My question is, does his proof limit the possibility of these pairs of primes to only one pair, or does it potentially allow for all the 7 million possibilities? Thanks! By the way
Cites Arthur · 2016-12 · Mathematics Stack Exchange · on “Arbitrarily big prime gap”
@barakmanos In fact, until a few years ago, we didn't know whether there were _any_ $n$ such that there are infinitely many consecutive prime pairs $p,q$ with $p-q=n$. We now know that the smallest such $n$ is below 70 million ( [this video] ) (There might've been improvements on this result since.)
Cites Arthur · 2017-12 · Mathematics Stack Exchange · on “Infinitely many pairs of sexy primes?”
We know that there is at least one difference smaller than seventy million [this video] (this result may have been improved since) which gives infinitely many pairs of primes. Whether $6$ is one of them we don't know.
Cites Mr Pie · 2018-08 · Mathematics Stack Exchange · on “Twin, cousin, sexy, ... primes”
@Babelfish There is a booked called *Things to Make and Do in the Fourth Dimension* (2014) written by Australian stand-up comedian and mathematician (and member of the Numberphile [LINK] crew) who talked about the polymath project in his book (Chapter Seven: *Prime Time*, page $152$). He wrote that as of $20$ July, $2013$, the brought the gap from just under " $70$ Million [this video] " to $5,414$. I then saw on wikipedia that they proved there are infinitely many primes with a gap of $246$.