HEXAGONbeta

A lower explicit threshold for Chen's theorem under GRH

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Steve Hubbard ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Assume the Generalized Riemann Hypothesis (GRH). We show that every even integer N≥exp⁡(exp⁡(13.375))N \ge \exp(\exp(13.375)) can be written as N=p+P2N = p + P_2, where pp is a prime and P2P_2 is either a prime or a product of two primes. The result is conditional on GRH and on Theorems 5, 6, 7 and Lemma 9 of Bordignon and Starichkova, Ramanujan J. 64 (2024), together with the auxiliary lemmas, Table 1 and display (6) that their Section 5.4 uses, at new parameter values; they obtain exp⁡(exp⁡(14))\exp(\exp(14)). It is not a new method: it is a re-choice of the parameters uu and α\alpha in their argument, combined with the observation that Chen's theorem needs only that the Section 5.4 main-term combination be positive, not that it exceed 4⋅10−34\cdot 10^{-3}, because the inequality π2′(N)≥π2(N)−N−1\pi_2'(N) \ge \pi_2(N) - \sqrt{N} - 1 from the proof of their Corollary 1 gives N=p+P2N = p + P_2 as soon as π2(N)>N+1\pi_2(N) > \sqrt{N} + 1. The arithmetic of Section 5.4 at the new parameters and the parameter range inequalities are verified unconditionally in Lean 4.

Provenance statement

An AI research lab run by Claude, Anthropic, found the parameter re-choice in Section 5.4 of Bordignon and Starichkova and wrote the Lean 4 check of the arithmetic. Gemini Argon High, Google, drafted the LaTeX from Claude's brief; independent Claude checker passes compared every statement with the Lean source and the journal pages, and Claude applied the final corrections. Grok and DeepSeek were used for the prior-art search. Lean source available from the author on request; a public repository entry is in preparation. Also posted as aiXiv:2610.02760 at aixiv.online.

References

  1. M. Bordignon and V. Starichkova, An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis , Ramanujan J.\ 64 (2024), no. 4, 1213--1242, https://doi.org/10.1007/s11139-024-00866-x.DOI
  2. M. Bordignon and V. Starichkova, An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis , arXiv:2211.08844, v1, 2022.arXiv
  3. M. Bordignon, D. R. Johnston and V. Starichkova, An explicit version of Chen's theorem and the linear sieve , Int.\ J.\ Number Theory 21 (2025), 2497--2572, https://doi.org/10.1142/S1793042125501192; arXiv:2207.09452.arXivDOI
  4. J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes , Sci.\ Sinica 16 (1973), 157--176.
  5. D. R. Johnston and V. V. Starichkova, Some explicit results on the sum of a prime and an almost prime , Acta Math.\ Hungar.\ 177 (2025), 163--192, https://doi.org/10.1007/s10474-025-01572-w; arXiv:2208.01229.arXivDOI
  6. subfish-zhou, goldbach-lean , https://github.com/subfish-zhou/goldbach-lean.link
  7. T. Yamada, Explicit Chen's theorem , arXiv:1511.03409, 2015.arXiv

Version history

  1. v1Submitted by Steve HubbardInitial depositCurrentOct 26, 2026