# Yitang Zhang: Prime Gaps

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?

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No doubt, Yitang Zhang's life will change as a result! – amWhy May 22 '13 at 1:18
I love that he's from a little known place. – Trancot May 22 '13 at 1:22
@Trancot, and over 50. – lhf May 22 '13 at 1:23
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 1:48
From Tao's G+ feed: plus.google.com/114134834346472219368/posts/39tuzQ8npYt. The current bound is 59,470,640. – Andrés E. Caicedo May 30 '13 at 17:51

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.

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That is so odd. Well, I mean it's even, but it is strange. – Trancot May 22 '13 at 2:14
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 michaelnielsen.org/polymath1/…. – lhf Jul 17 '13 at 4:33
Both links appear broken... – barto Mar 10 '15 at 20:00
@barto, fixed, thanks. – lhf Mar 11 '15 at 4:38

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 $63, 374, 610.$

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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

It's now down to 600.

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That post actually says it's down to 246. – Micah Mar 11 '15 at 5:19