Yitang Zhang: Prime Gaps

Question

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific.

*EDIT*$^1$:

Are there any experts here who can explain the proof? Is the outline in the annals the preprint or the full accepted paper?

I find it really inspiring...kind of an "in your face, so there" thang! — amWhy, May 22 '13 at 01:24
You can download a copy of the full accepted paper on the Annals page if your institution subscribes to the Annals. — lhf, May 22 '13 at 01:48
@lhf What database would that be available through? Ebsco? Jstor? AMS? MathSci? — Trancot, May 22 '13 at 01:57
Ah, I didn't know that Annals allows early access. Indeed, I have done as lhf has mentioned and examined (briefly) the paper (which is not to say that I can now explain it). Thanks @lhf — davidlowryduda, May 22 '13 at 02:23
From Tao's G+ feed: https://plus.google.com/114134834346472219368/posts/39tuzQ8npYt. The current bound is 59,470,640. — Andrés E. Caicedo, May 30 '13 at 17:51
Now it is 48,112,378. See http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ and the comments there. — Andrés E. Caicedo, Jun 01 '13 at 05:41
As spectator sports go, this is ridiculously engaging. The bound is now at most 388,248. — Andrés E. Caicedo, Jun 05 '13 at 21:44
For an up-to-date table of "records", with references, see column 4 of the table at http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records — Andrés E. Caicedo, Jun 05 '13 at 21:54
A paper is currently being written as a result of the Polymath project. The value they stopped at is 4,680. https://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ — Andrés E. Caicedo, Aug 31 '13 at 02:50
GOOD FOR HIM! AND YET ANOTHER COUNTEREXAMPLE TO THE OLD "SCIENCE IS A GAME FOR THE YOUNG,OVER 30,YOU'RE FINISHED" MALARKY! — Mathemagician1234, Sep 20 '13 at 00:07
I obtained a copy of the paper at here, for anyone of potential interest. — Bombyx mori, May 22 '13 at 07:31
@Mathemagician1234: Turn your hearing aid up, grandpa... you're yelling again. :) — mjqxxxx, Mar 11 '15 at 05:07

lhf · Accepted Answer · 2015-03-11T04:38:04.750

22

70 million is exactly what is mentioned in the abstract.

It is quite likely that this bound can be reduced; the author says so in the paper:

This result is, of course, not optimal. The condition $k_0 \ge 3.5 \times 10^6$ is also crude and there are certain ways to relax it. To replace the right side of (1.5) by a value as small as possible is an open problem that will not be discussed in this paper.

He seems to be holding his cards for the moment...

You can download a copy of the full accepted paper on the Annals page if your institution subscribes to the Annals.

edited Mar 11 '15 at 04:38

answered May 22 '13 at 01:11

lhf

221,500

21

That is so odd. Well, I mean it's even, but it is strange. – Trancot May 22 '13 at 02:14
For one, the paper does actually say that the result is not optimal. For two, the limit inferior must be even, as you are taking the difference of two odd numbers (consecutive primes). This result also does not say much about related conjectures: the Hardy-Littlewood k-tuple conjecture being probably the most important. It is my belief that a proof of the twin prime conjecture, done right, will also prove most, if not all, of the related conjectures. – Adam Jun 23 '13 at 17:16
The original bound has been drastically reduced. See the $H$ column in http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records. – lhf Jul 17 '13 at 04:33
Both links appear broken... – Bart Michels Mar 10 '15 at 20:00
@barto, fixed, thanks. – lhf Mar 11 '15 at 04:38

score 7 · Answer 2 · edited Apr 13 '17 at 12:58

7

As to the "idea" of the proof, I would suggest looking at the following.

As mentioned by Mark Lewko, according to WolframAlpha, the proof gives a gap size of $63, 374, 610.$

edited Apr 13 '17 at 12:58

Community

1

answered May 22 '13 at 07:17

leshik

4,990

5

Please try to describe as much here as possible in order to make the answer self-contained. Links are fine as support, but they can go stale and then an answer which is nothing more than a link loses its value. – robjohn May 22 '13 at 13:30

score 1 · Answer 3 · answered Mar 11 '15 at 05:14

1

It's now down to 600.

See here: https://cameroncounts.wordpress.com/2015/01/22/polymath/

answered Mar 11 '15 at 05:14

marty cohen

110,450

That post actually says it's down to 246. – Micah Mar 11 '15 at 05:19

Yitang Zhang: Prime Gaps

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