Representing an integer and its powers in two unrelated number systems

[2502.00296] Representing an integer and its powers in two unrelated number systems Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2502.00296 (math) [Submitted on 1 Feb 2025 ( v1 ), last revised 11 Apr 2026 (this version, v2)] Title: Representing an integer and its powers in two unrelated number systems Authors: Divyum Sharma , L. Singhal View a PDF of the paper titled Representing an integer and its powers in two unrelated number systems, by Divyum Sharma and L. Singhal View PDF HTML (experimental) Abstract: Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent denominators to $\alpha$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca. Comments: Modified version to appear in Acta Arithmetica Subjects: Number Theory (math.NT) MSC classes: 11D61, 11J86, 11A63, 11B39 Cite as: arXiv:2502.00296 [math.NT] (or arXiv:2502.00296v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2502.00296 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Acta Arith. 223 (2026) 309-338 Related DOI : https://doi.org/10.4064/aa250218-13-4 Focus to learn more DOI(s) linking to related resources Submission history From: L. Singhal [ view email ] [v1] Sat, 1 Feb 2025 03:38:27 UTC (17 KB) [v2] Sat, 11 Apr 2026 05:58:22 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled Representing an integer and its powers in two unrelated number systems, by Divyum Sharma and L. Singhal View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-02 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

Related documents

• The difficulty of prime factorization is a consequence of the positional numeral system#766890
• On the multiplicative representation of integers#490165
• The number-system of algebra treated theoretically and historically (2nd edition)#39285
• Multiplicative number theory. I. Classical theory#490196
• Unrestricted division paradox and the contingency of division#96774