math.NT — Number Theory

Improved bounds on $H_m$ for $2\le m\le 5$

Contributed by Annanay Kapila

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 preprint proving $H_1\le186$. Together with a criterion from the same preprint for the triple dense divisibility of least common multiples, it allows the sieve weights to be supported, instead of on Polymath's truncated simplex, on a region in which single coordinates may be as large as about $0.38$. On this region we use product trial functions multiplied by an optimised function of the sum of the coordinates, and we certify the resulting variational quotients in ball arithmetic, computing the relevant distribution functions by Fourier inversion.