Improved bounds on $H_m$ for $2\le m\le 5$
Let $H_m=\liminf_{n\to\infty}(p_{n+m}-p_n)$, where $p_n$ denotes the $n$th prime. We prove \[ H_2\le 20{,}370,\qquad H_3\le 1{,}380{,}156,\qquad H_4\le 82{,}959{,}170,\qquad H_5\le 5{,}040{,}082{,}836, \] improving the bounds $33{,}118$, $2{,}718{,}108$, $214{,}099{,}720$ and $14{,}541{,}349{,}288$ of Roeling (2026). We use the multidimensional Selberg sieve of Maynard and Tao in the form developed by Polymath. The arithmetic input is Stadlmann's equidistribution estimate for primes in arithmetic progressions, in the extension to triply densely divisible moduli given in OpenAI's 2026…
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