I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart

Everyone by now has heard about Zhang’s landmark result showing that there are infinitely many pairs of primes at most 70000000 apart.

His core result is that if a set of 3.5 * 10^6 (corrected, thanks to comment #2) numbers $H$ is admissible (see below), then there are infinitely many $n$ so that $n+H$ contains at least two primes. He then easily constructs an admissible set wherein the largest difference is 7 * 10^7, obtaining the stated result.

A set $H$ is admissible if there is no prime $p$ so $H \pmod p$ occupies every residue class. For a given $H$ this is clearly a checkable condition; there’s no need to look at primes larger than $|H|$.

(While Zhang went for a nice round number, Mark Lewko found his argument in fact gives 63374611, if you’re being overly specific about these things, which we are right now. :-)

In a short note on the arXiv yesterday, Tim Trudgian (whose office is not far from mine) pointed out another way to build an admissible set, giving a smaller largest difference, obtaining the result that there are infinitely many pairs of primes at most 59874594 apart. He considers sets of the form $H_m = {p_{m+1}, \ldots, p_{m+k_0}}$ (where $k_0$ is Zhang’s constant 3.5 * 10^7). These aren’t necessarily admissible, but they are for some values of $m$, and both Zhang and Tim noticed certain values for which this is easy to prove. Zhang used $H_m$ with $m=k_0$, while Tim’s observation is that $m_0=250150=\pi(k_0)$ also works. (Comment #2 below points out this isn’t right, and Zhang also intended $m=\pi(k_0)$, and the slack in his estimate is coming from just looking at the largest element of $H_m$, rather than the largest difference.) Thus the bound in his result is $p_{m_0+k_0}-p_{m_0+1} = 59874594$.

It turns out that checking admissibility for a given $H_m$ isn’t that hard; it takes about an hour to check a single value for $m ~ m_0$ (but if you find a prime witnessing $H_m$ not being admissible, it very often gives you a fast proof that $H_{m+1}$ is not admissible either, so searching is much faster).

I haven’t looked exhaustively, but one can check that $m_1 = m_0 / 2 = 125075$ gives an admissible $H_m$, and hence there are infinitely many pairs of primes at most $p_{m_1 + k_0} - p_{m_1 + 1} = 59470640$. (Sadly, it’s impossible to get below 59 million with this trick; no $m$ below 27000 works; all witnessed by $p=182887$ or $378071$.)

I just couldn’t resist momentarily “claiming the crown” for the smallest upper bound on gap size. :-) Of course the actual progress, that’s surely coming soon from people who actually understand Zhang’s work, is going to be in reducing his 3.5 * 10^6. You can read more about prospects for that in the answers to this MathOverflow question.