{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T17:24:05Z","timestamp":1779211445175,"version":"3.51.4"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T00:00:00Z","timestamp":1746144000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T00:00:00Z","timestamp":1746144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Brno University of Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Supercomput"],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This article presents our project, which aims to verify the Collatz conjecture computationally. As a main point of the article, we introduce a new result that pushes the limit for which the conjecture is verified up to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$2^{71}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mn>71<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. We present our baseline algorithm and then several sub-algorithms that enhance acceleration. The total acceleration from the first algorithm we used on the CPU to our best algorithm on the GPU is <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1\\,335\\times$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mspace\/>\n                    <mml:mn>335<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. We further distribute individual tasks to thousands of parallel workers running on several European supercomputers. Besides the convergence verification, our program also checks for path records during the convergence test. We found four new path records.<\/jats:p>","DOI":"10.1007\/s11227-025-07337-0","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T12:42:42Z","timestamp":1746189762000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Improved verification limit for the convergence of the Collatz conjecture"],"prefix":"10.1007","volume":"81","author":[{"given":"David","family":"Barina","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,2]]},"reference":[{"key":"7337_CR1","doi-asserted-by":"crossref","unstructured":"Lagarias JC (ed) (2010) The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society","DOI":"10.1090\/mbk\/078"},{"key":"7337_CR2","unstructured":"Lagarias JC (2003) The $$3x+1$$ problem: An annotated bibliography (1963\u20131999) (sorted by author). arXiv:math\/0309224"},{"key":"7337_CR3","unstructured":"Lagarias JC (2006) The $$3x+1$$ problem: An annotated bibliography, II (2000-2009). arXiv:math\/0608208"},{"issue":"3","key":"7337_CR4","doi-asserted-by":"publisher","first-page":"2681","DOI":"10.1007\/s11227-020-03368-x","volume":"77","author":"D Barina","year":"2021","unstructured":"Barina D (2021) Convergence verification of the Collatz problem. J Super Comput 77(3):2681\u20132688. https:\/\/doi.org\/10.1007\/s11227-020-03368-x","journal-title":"J Super Comput"},{"key":"7337_CR5","volume-title":"On Ulam\u2019s Problem","author":"R Dunn","year":"1973","unstructured":"Dunn R (1973) On Ulam\u2019s Problem. University of Colorado at Boulder, Tech. rep"},{"issue":"11","key":"7337_CR6","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0898-1221(92)90034-F","volume":"24","author":"GT Leavens","year":"1992","unstructured":"Leavens GT, Vermeulen M (1992) 3x+1 Search programs. Comput Math Appl 24(11):79\u201399. https:\/\/doi.org\/10.1016\/0898-1221(92)90034-F","journal-title":"Comput Math Appl"},{"key":"7337_CR7","unstructured":"Roosendaal E (2019) personal communication"},{"key":"7337_CR8","unstructured":"Oliveira e Silva T (2010) Empirical verification of the 3x+1 and related conjectures. In: Lagarias JC (ed) The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society, pp 189\u2013207"},{"key":"7337_CR9","unstructured":"Hercher C (2018) \u00dcber die L\u00e4nge nicht-trivialer Collatz-Zyklen. Die Wurzel 6 and 7"},{"issue":"1","key":"7337_CR10","first-page":"69","volume":"7","author":"T Honda","year":"2017","unstructured":"Honda T, Ito Y, Nakano K (2017) GPU-accelerated exhaustive verification of the Collatz conjecture. Int J Netw Comput 7(1):69\u201385","journal-title":"Int J Netw Comput"},{"issue":"225","key":"7337_CR11","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1090\/S0025-5718-99-01031-5","volume":"68","author":"TOE Silva","year":"1999","unstructured":"Silva TOE (1999) Maximum excursion and stopping time record-holders for the $$3x+1$$ problem: computational results. Math Comput 68(225):371\u2013384. https:\/\/doi.org\/10.1090\/S0025-5718-99-01031-5","journal-title":"Math Comput"},{"issue":"3","key":"7337_CR12","first-page":"1","volume":"26","author":"C Hercher","year":"2023","unstructured":"Hercher C (2023) There are no Collatz $$m$$-cycles with $$m \\le 91$$. J Integer Seq 26(3):1\u201322","journal-title":"J Integer Seq"},{"issue":"1","key":"7337_CR13","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1214\/aoap\/1177005779","volume":"2","author":"JC Lagarias","year":"1992","unstructured":"Lagarias JC, Weiss A (1992) The $$3x + 1$$ problem: two stochastic models. Annals Appl Probab 2(1):229\u2013261. https:\/\/doi.org\/10.1214\/aoap\/1177005779","journal-title":"Annals Appl Probab"}],"container-title":["The Journal of Supercomputing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-025-07337-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11227-025-07337-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-025-07337-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T12:42:47Z","timestamp":1746189767000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11227-025-07337-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,2]]},"references-count":13,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["7337"],"URL":"https:\/\/doi.org\/10.1007\/s11227-025-07337-0","relation":{},"ISSN":["1573-0484"],"issn-type":[{"value":"1573-0484","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,2]]},"assertion":[{"value":"21 April 2025","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2025","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"810"}}