{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/algorithms-for-the-multiplication-table","title":"Algorithms for the Multiplication Table Problem","arxiv_id":"1908.04251","date":"2019-08-12","proceeding":null,"authors":["Richard Brent","Carl Pomerance","David Purdum","Jonathan Webster"],"abstract":"Let $M(n)$ denote the number of distinct entries in the $n \\times n$ multiplication table. The function $M(n)$ has been studied by Erd\\H{o}s, Tenenbaum, Ford, and others, but the asymptotic behaviour of $M(n)$ as $n \\to \\infty$ is not known precisely. Thus, there is some interest in algorithms for computing $M(n)$ either exactly or approximately. We compare several algorithms for computing $M(n)$ exactly, and give a new algorithm that has a subquadratic running time. We also present two Monte Carlo algorithms for approximate computation of $M(n)$. We give the results of exact computations for values of $n$ up to $2^{30}$, and of Monte Carlo computations for $n$ up to $2^{100,000,000}$, and compare our experimental results with Ford's order-of-magnitude result.","url_abs":"https://arxiv.org/abs/1908.04251v2","url_pdf":"https://arxiv.org/pdf/1908.04251v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"algorithms-for-the-multiplication-table","repo_url":"https://github.com/rutrum/multiplication-table","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}