[2305.06306] Random Diophantine Equations in the Primes Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2305.06306 (math) [Submitted on 10 May 2023 ( v1 ), last revised 28 Apr 2026 (this version, v2)] Title: Random Diophantine Equations in the Primes Authors: Philippa Holdridge View a PDF of the paper titled Random Diophantine Equations in the Primes, by Philippa Holdridge View PDF HTML (experimental) Abstract: We consider equations of the form $a_{1}x_{1}^{k}+...+a_{s}x_{s}^{k}$ and when they have solutions in the primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove that, whenever $s\ge 3k+2$, this holds for almost all such equations. This is based on work of Brüdern and Dietmann on the Hasse principle. We then prove some further results about prime solubility and the prime Hasse principle, including a partial converse, and some counterexamples. Of particular interest are counterexamples of degree 2, which show that the analogue of the Hasse-Minkowski theorem fails for prime solubility. Comments: 52 pages, to appear in Mathematika Subjects: Number Theory (math.NT) MSC classes: 11P55, 11D72, 11P32, 11E76 Cite as: arXiv:2305.06306 [math.NT] (or arXiv:2305.06306v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2305.06306 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Philippa Holdridge [ view email ] [v1] Wed, 10 May 2023 16:51:42 UTC (38 KB) [v2] Tue, 28 Apr 2026 07:35:54 UTC (39 KB) Full-text links: Access Paper: View a PDF of the paper titled Random Diophantine Equations in the Primes, by Philippa Holdridge View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2023-05 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