The Exceptional Set in Goldbach's Problem with two Chen Primes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909749197406208 |
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| author | Grimmelt, Lasse Teräväinen, Joni |
| author_facet | Grimmelt, Lasse Teräväinen, Joni |
| contents | We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is optimal, barring substantial progress on the twin prime or binary Goldbach conjectures.
The proof is based on constructing a non-negative model for the Chen primes in a suitable approximate sense. To do this, we develop an efficient sieving strategy that makes use of a power-saving variant of the Bombieri--Vinogradov theorem. Furthermore, we show that the primes are well approximated in additive problems by the Cramér model (rough numbers) with a sifting parameter of power size. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16400 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Exceptional Set in Goldbach's Problem with two Chen Primes Grimmelt, Lasse Teräväinen, Joni Number Theory We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is optimal, barring substantial progress on the twin prime or binary Goldbach conjectures. The proof is based on constructing a non-negative model for the Chen primes in a suitable approximate sense. To do this, we develop an efficient sieving strategy that makes use of a power-saving variant of the Bombieri--Vinogradov theorem. Furthermore, we show that the primes are well approximated in additive problems by the Cramér model (rough numbers) with a sifting parameter of power size. |
| title | The Exceptional Set in Goldbach's Problem with two Chen Primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.16400 |