On Frobenius Numbers of Shifted Power Sequences

[2210.02722] On Frobenius Numbers of Shifted Power Sequences Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2210.02722 (math) [Submitted on 6 Oct 2022 ( v1 ), last revised 23 Jul 2025 (this version, v4)] Title: On Frobenius Numbers of Shifted Power Sequences Authors: Feihu Liu , Guoce Xin View a PDF of the paper titled On Frobenius Numbers of Shifted Power Sequences, by Feihu Liu and Guoce Xin View PDF HTML (experimental) Abstract: We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$. Comments: 16 pages 2 tables Subjects: Combinatorics (math.CO) MSC classes: 11D07, 05A15, 11B75, 11D04 Cite as: arXiv:2210.02722 [math.CO] (or arXiv:2210.02722v4 [math.CO] for this version) https://doi.org/10.48550/arXiv.2210.02722 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s00026-026-00811-3 Focus to learn more DOI(s) linking to related resources Submission history From: FeiHu Liu [ view email ] [v1] Thu, 6 Oct 2022 07:23:00 UTC (24 KB) [v2] Mon, 27 Mar 2023 12:40:56 UTC (22 KB) [v3] Wed, 26 Jun 2024 08:12:02 UTC (17 KB) [v4] Wed, 23 Jul 2025 03:21:13 UTC (17 KB) Full-text links: Access Paper: View a PDF of the paper titled On Frobenius Numbers of Shifted Power Sequences, by Feihu Liu and Guoce Xin View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2022-10 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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