An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime

[2608.02381] An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2608.02381 (math) [Submitted on 3 Aug 2026 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime Authors: Tushar Pandey View PDF HTML (experimental) Abstract: We study the statement that every even number $n \ge 6$ is the sum of a prime and a member of a twin prime pair. It sits between the conjectures of Goldbach and Dubner and implies both the Goldbach and the twin prime conjectures. Our main result is conditional: if the number of twin primes up to $z$ is at least $c\,z/\log^2 z$ for all large $z$, a lower bound of the order predicted by Hardy and Littlewood, with nothing assumed about their distribution, then a positive proportion of the even numbers are so representable, with density at least an absolute multiple of $c$. The proof is Romanov's method with a Selberg sieve, and loses no factor of $\log\log$. Nothing in this direction can be unconditional, since representability on a set of positive density already implies the twin prime conjecture; we show further that a polylogarithmic bound on the least twin summand would force a power-type lower bound on the number of twin primes. We verify the statement exhaustively for all even numbers up to $10^{14}$, where the least twin summand never exceeds 23,029. Comments: 15 pages. v2: verification extended to 10^14; conditional Romanov-type density theorem added; references updated. Code and data: doi: https://doi.org/10.5281/zenodo.21744861 Subjects: Number Theory (math.NT) MSC classes: 11P32, 11N36, 11Y11 Cite as: arXiv:2608.02381 [math.NT] (or arXiv:2608.02381v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.02381 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Tushar Pandey [ view email ] [v1] Mon, 3 Aug 2026 15:25:12 UTC (13 KB) [v2] Wed, 26 Aug 2026 20:31:46 UTC (22 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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