4 ms·

The Beauty of Bounded Gaps (2013)

44 points by cyang08 10y ago

aisofteng 10y ago
This is from 2013.
- danielmorozoff 10y ago
  And there has been much progress made in reducing the gap. Check out the polymath project and work done in collaboration with terry tao
- pmalynin 10y ago
  Yup, I've been taught the result in first year calculus and the resulting corollaries, which was over 2 years ago now.
  - JadeNB 10y ago
    > Yup, I've been taught the result in first year calculus and the resulting corollaries, which was over 2 years ago now. Are you joking? This is not calculus, and it's certainly not first-year material.
    - williamstein 10y ago
      I think he means that he stated the result in a calc class, not that he proved it. It would be very reasonable (and cool!) to state without proof in a calc class.
      - JadeNB 10y ago
        I certainly won't presume to tell Will Stein about number theory, but it seems strange to me to include in a calculus class (even without proof). Far be it from me to suggest ever excluding interesting mathematical content, but I'd be suspicious of an ordinary calculus class appreciating the significance of this result. On the other hand, maybe such prophecies are self fulfilling, and the enthusiasm of the teacher engenders the enthusiasm of his or her students, whatever the material. (I am curious what "the resulting corollaries" are, though.)
        pmalynin 10y ago
        Hmm, well you can read for yourself, my prof is nice enough to have posted all of the lecture notes online; These are the lectures from the second class where the result is covered on the third page: https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140904.pdf https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090... And then the homework (Q3): https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140904HW1.pdf https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090... Where the corollary follows.
        JadeNB 10y ago
        To be clear, I didn't mean "Are you joking?" as in "You are obviously lying or mistaken"; I really meant, exactly and only, "Are you joking?", because I found it incredible. Well, obviously you are not; and I would have taken you at your word without proof, but thank you for the careful documentation. I still would be hesitant to talk about this in my own calculus courses, but, given the impression that this made on you and Will Stein's endorsement (https://news.ycombinator.com/item?id=12886751 https://news.ycombinator.com/item?id=12886751), it sounds like that's just undue timidity on my part.
    - jvln 10y ago
      It is a subject of analytic number theory. Analytic stands for complex variable calculus.
JadeNB 10y ago
If you're a fan of "How not to be wrong" (http://www.jordanellenberg.com/how-not-to-be-wrong http://www.jordanellenberg.com/how-not-to-be-wrong )—and, if you're not, then go read it and you probably will be—then it may be worth noting that this article is by the same author. It is an excellent and accessible exposition that doesn't shy away from pointing to more technical resources for those who are interested.
kazinator 10y ago
> Quite the opposite—we take [primes] as immutable features of the universe, and carve them on the golden records we shoot out into interstellar space to prove to the ETs that we’re no dopes. That sentence itself is golden!
dng88 10y ago
"When you divide a big prime number by 3, the quotient can’t come out even" should be zero not even. A very good article on prime no nevertheless !
duerrp 10y ago
There's a nice documentary on Zhang (and the progress his work triggered) here: https://www.youtube.com/watch?v=XIIyKWxGhEA https://www.youtube.com/watch?v=XIIyKWxGhEA

44 points by cyang08 10y ago

aisofteng 10y ago

This is from 2013.

danielmorozoff 10y ago
And there has been much progress made in reducing the gap. Check out the polymath project and work done in collaboration with terry tao
pmalynin 10y ago
Yup, I've been taught the result in first year calculus and the resulting corollaries, which was over 2 years ago now.
- JadeNB 10y ago
  > Yup, I've been taught the result in first year calculus and the resulting corollaries, which was over 2 years ago now. Are you joking? This is not calculus, and it's certainly not first-year material.
  - williamstein 10y ago
    I think he means that he stated the result in a calc class, not that he proved it. It would be very reasonable (and cool!) to state without proof in a calc class.
    - JadeNB 10y ago
      I certainly won't presume to tell Will Stein about number theory, but it seems strange to me to include in a calculus class (even without proof). Far be it from me to suggest ever excluding interesting mathematical content, but I'd be suspicious of an ordinary calculus class appreciating the significance of this result. On the other hand, maybe such prophecies are self fulfilling, and the enthusiasm of the teacher engenders the enthusiasm of his or her students, whatever the material. (I am curious what "the resulting corollaries" are, though.)
      - pmalynin 10y ago
        Hmm, well you can read for yourself, my prof is nice enough to have posted all of the lecture notes online; These are the lectures from the second class where the result is covered on the third page: https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140904.pdf https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090... And then the homework (Q3): https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140904HW1.pdf https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090... Where the corollary follows.
        JadeNB 10y ago
        To be clear, I didn't mean "Are you joking?" as in "You are obviously lying or mistaken"; I really meant, exactly and only, "Are you joking?", because I found it incredible. Well, obviously you are not; and I would have taken you at your word without proof, but thank you for the careful documentation. I still would be hesitant to talk about this in my own calculus courses, but, given the impression that this made on you and Will Stein's endorsement (https://news.ycombinator.com/item?id=12886751 https://news.ycombinator.com/item?id=12886751), it sounds like that's just undue timidity on my part.
  - jvln 10y ago
    It is a subject of analytic number theory. Analytic stands for complex variable calculus.

JadeNB 10y ago

If you're a fan of "How not to be wrong" (http://www.jordanellenberg.com/how-not-to-be-wrong http://www.jordanellenberg.com/how-not-to-be-wrong )—and, if you're not, then go read it and you probably will be—then it may be worth noting that this article is by the same author. It is an excellent and accessible exposition that doesn't shy away from pointing to more technical resources for those who are interested.

kazinator 10y ago

> Quite the opposite—we take [primes] as immutable features of the universe, and carve them on the golden records we shoot out into interstellar space to prove to the ETs that we’re no dopes. That sentence itself is golden!

dng88 10y ago

"When you divide a big prime number by 3, the quotient can’t come out even" should be zero not even. A very good article on prime no nevertheless !

duerrp 10y ago

There's a nice documentary on Zhang (and the progress his work triggered) here: https://www.youtube.com/watch?v=XIIyKWxGhEA https://www.youtube.com/watch?v=XIIyKWxGhEA