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on Bounded prime gap constant ($C_{88a}$)

The statement

Let $p_n$ denote the $n$ -th prime. The bounded prime gap constant is

C_{88a} \;=\; H_1 \;:=\; \liminf_{n \to \infty} (p_{n+1} - p_n),

the least limit point of the sequence of gaps between consecutive primes. That $H_1$ is finite — that some bounded gap recurs infinitely often — was proved by Zhang [Zha14] in 2013; the twin prime conjecture is exactly the assertion $H_1 = 2$ .

More generally one writes $H_m := \liminf_{n\to\infty}(p_{n+m} - p_n)$ for $m \ge 1$ . Maynard [May15] proved $H_m$ finite for every $m$ ; the growth rate of $H_m$ in $m$ is recorded separately as $C_{88b}$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.