A repository has been released concerning the target bound that there exist infinitely many pairs of consecutive primes in the sequence of prime numbers p_n with a maximum gap of 186.
Repository Released for Formalization and Numerical Proof of Prime Gap Intervals Using Lean 4
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In this development, DHL[40,2] has been derived based on the condition that there exist infinitely many transformations containing at least two primes in all allowable sets including 40 integer shifts.
The deliverables include formalization using Lean 4 and numerical proofs using Python.
The results via Lean depend on three explicit input axioms. These mathematical estimates and numerical calculations do not constitute proofs by Lean itself and are treated as inputs.
The axioms used include bounds regarding Kloosterman sums. These are defined based on existing literature such as Deligne's theorem and works by Étienne Fouvry et al.
Python 3.12.13 was used for the numerical proofs.
Calculations were recalculated from scratch using libraries such as NumPy and python-flint. Note that these numerical proofs do not serve as a substitute for the Lean axioms as proofs.
--- Sources: GPT-6-Astra: infinitely pairs of consecutive primes with distance at most 186 (Hacker News Frontpage, 2026-09-04)