{"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/an-algorithm-and-computation-to-verify","title":"An algorithm and computation to verify Legendre's Conjecture up to $3.33\\cdot10^{13}$","arxiv_id":"2401.13753","date":"2024-01-24","proceeding":null,"authors":["Jonathan Sorenson","Jonathan Webster"],"abstract":"We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre's conjecture claims that for every positive integer $n$, there exists a prime between $n^2$ and $(n+1)^2$. Oppermann's conjecture subsumes Legendre's conjecture by claiming there are primes between $n^2$ and $n(n+1)$ and also between $n(n+1)$ and $(n+1)^2$. Using Cram\\'er's conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann's conjecture, and hence also Legendre's conjecture, for all $n\\le N$ in time $O( N \\log N \\log \\log N)$ and space $N^{O(1/\\log \\log N)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann's conjecture from the previous $N = 2\\cdot 10^{9}$ up to $N = 3.33\\cdot 10^{13}$, so we were finding $27$ digit primes. The computation ran for about half a year on four Intel Xeon Phi $7210$ processors using a total of $256$ cores.","url_abs":"https://arxiv.org/abs/2401.13753v1","url_pdf":"https://arxiv.org/pdf/2401.13753v1.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":"an-algorithm-and-computation-to-verify","repo_url":"https://github.com/sorenson64/olc","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}