Explicit bounds for H_3 and H_4 from checked admissible tuples
Year: 2026
Type: preprint
Abstract: Let H_m = liminf_{n -> infinity} (p_{n+m} - p_n), where p_n is the n-th prime. I prove H_3 <= 24,354,332 and H_4 <= 1,390,484,966. The proof has three ingredients. The published implications are Stadlmann's distribution estimate (Corollary 2 of arXiv:2309.00425v3) and Theorems 3.10 and 6.7 and Claim 3.1 of Polymath8b. An evaluation of the lower bound of Polymath8b Theorem 6.7 in rational interval arithmetic, at two exact rational points, gives DHL[1,624,333, 4] and DHL[73,796,369, 5]. Two admissible tuples, of 1,624,333 and 73,796,369 entries, have diameters 24,354,332 and 1,390,484,966, and two independent programs checked each of them. The two bounds improve the explicit bounds 24,407,016 and 1,391,051,532 of Stadlmann's Corollary 1. The tuples, the sieve certificate of the H_4 tuple, the checkers and the evaluator with its outputs are deposited with this note. Version 2 replaces the H_4 tuple of version 1 by a narrower one from a grid over the sieve parameters; the chain is otherwise unchanged. Built and tested with Claude Opus 5.5.
Author: Ayub, Amir
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225377861