Almost all primes are partially regular

These files accompany the paper arXiv:2602.05090.

The formal proofs provided in this work were developed and verified using Lean 4.26.0. Compatibility with earlier or later versions is not guaranteed due to the evolving nature of the Lean 4 compiler and its core libraries.

Input files

• problem.tex: natural language description of the problem
• .environment: specifies the Lean version

Output files (Run with Lean 4.26.0)

• PartialRegularity/problem.lean: translation of the problem statement into formal language (Lean)
• PartialRegularity/solution.lean: solution in formal language (Lean)

Additional files

Besides the end-to-end run above, we internally also conducted experiments where we provided AxiomProver with additional tools, such as AlexKontorovich/PrimeNumberTheoremAnd (although this dependency ended up not being needed). In one of these experiments, AxiomProver found that actually $C_{\alpha} = 10$ works for all $\alpha > 1/2$.

We provide the corresponding Lean files in extension/.

Verifying with Comparator

This repository can be verified against the formal problem statement with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:

lake env comparator comparator.json
lake env comparator comparator-extension.json

License

This repository uses the MIT License. See LICENSE for details.

Repository maintainers

• Evan Chen
• Kenny Lau
• Seewoo Lee
• Ken Ono
• Jujian Zhang