Large prime factors of well-distributed sequences

[2402.11884] Large prime factors of well-distributed sequences Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2402.11884 (math) [Submitted on 19 Feb 2024 ( v1 ), last revised 9 Apr 2026 (this version, v4)] Title: Large prime factors of well-distributed sequences Authors: Abhishek Bharadwaj , Brad Rodgers View a PDF of the paper titled Large prime factors of well-distributed sequences, by Abhishek Bharadwaj and 1 other authors View PDF HTML (experimental) Abstract: We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of $u$ tend to a Poisson-Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson-Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of $u$ is greater than $u^{1-\epsilon}$, showing that this probability is $O(\epsilon)$. Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. The proofs involve (i) a characterization of the Poisson-Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve. Comments: 16 pages. Incorporates referee comments and corrections Subjects: Number Theory (math.NT) Cite as: arXiv:2402.11884 [math.NT] (or arXiv:2402.11884v4 [math.NT] for this version) https://doi.org/10.48550/arXiv.2402.11884 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Brad Rodgers [ view email ] [v1] Mon, 19 Feb 2024 06:46:27 UTC (13 KB) [v2] Wed, 10 Apr 2024 22:56:37 UTC (15 KB) [v3] Thu, 5 Jun 2025 19:08:25 UTC (16 KB) [v4] Thu, 9 Apr 2026 08:00:29 UTC (17 KB) Full-text links: Access Paper: View a PDF of the paper titled Large prime factors of well-distributed sequences, by Abhishek Bharadwaj and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-02 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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

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