{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:17:49Z","timestamp":1774437469785,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T00:00:00Z","timestamp":1561075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100018533","name":"Major Scientific and Technological Special Project of Guizhou Province","doi-asserted-by":"publisher","award":["20183001"],"award-info":[{"award-number":["20183001"]}],"id":[{"id":"10.13039\/501100018533","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Open Funding of Guizhou Provincial Key Laboratory of Public Big Data","award":["2018BDKFJJ009, 2017BDKFJJ006"],"award-info":[{"award-number":["2018BDKFJJ009, 2017BDKFJJ006"]}]},{"name":"Open Funding of Hubei Provincial Key Laboratory of Intelligent Geo-Information Processing","award":["KLIGIP2016A05"],"award-info":[{"award-number":["KLIGIP2016A05"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Data"],"abstract":"<jats:p>Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 \u00d7 x + 1 or x\/2 computations from a starting integer to the first integer smaller than the starting integer) and original dynamics of integers (from a starting integer to 1). Especially, the starting integer has no upper bound. That is, extremely large integers with length of about 100,000 bits, e.g., 2100000 \u2212 1, can be verified for Collatz conjecture, which is much larger than current upper bound (about 260). We analyze the properties of those data (e.g., reduced dynamics) and discover the following laws; reduced dynamics is periodic and the period is the length of its reduced dynamics; the count of x\/2 equals to minimal integer that is not less than the count of (3 \u00d7 x + 1)\/2 times ln(1.5)\/ln(2). Besides, we observe that all integers are partitioned regularly in half and half iteratively along with the prolonging of reduced dynamics, thus given a reduced dynamics we can compute a residue class that presents this reduced dynamics by a proposed algorithm. It creates one-to-one mapping between a reduced dynamics and a residue class. These observations from data can reveal the properties of reduced dynamics, which are proved mathematically in our other papers (see references). If it can be proved that every integer has reduced dynamics, then every integer will have original dynamics (i.e., Collatz conjecture will be true). The data set includes reduced dynamics of all odd positive integers in [3, 99999999] whose remainder is 3 when dividing 4, original dynamics of some extremely large integers, and all computer source codes in C that implement our proposed algorithms for generating data (i.e., reduced or original dynamics).<\/jats:p>","DOI":"10.3390\/data4020089","type":"journal-article","created":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T11:54:31Z","timestamp":1561118071000},"page":"89","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Reduced Collatz Dynamics Data Reveals Properties for the Future Proof of Collatz Conjecture"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8590-1737","authenticated-orcid":false,"given":"Wei","family":"Ren","sequence":"first","affiliation":[{"name":"School of Computer Science, China University of Geosciences, Wuhan 430079, China"},{"name":"Hubei Key Laboratory of Intelligent Geo-Information Processing, China University of Geosciences, Wuhan 430079, China"},{"name":"Guizhou Provincial Key Laboratory of Public Big Data, GuiZhou University, Guiyang 550025, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6129836","DOI":"10.1155\/2019\/6129836","article-title":"A New Approach on Proving Collatz Conjecture","volume":"2019","author":"Ren","year":"2019","journal-title":"J. Math."},{"key":"ref_2","unstructured":"Ren, W. (2019, May 01). Exploring Properties in Reduced Collatz Dynamics, IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/ge9h-8a77."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ren, W. (2019). A Reduced Collatz Dynamics Maps to a Residue Class, and its Count of x\/2 over Count of 3*x+1 is larger than ln3\/ln2. Math. Rep., submitted.","DOI":"10.36227\/techrxiv.11664567.v1"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ren, W. (2019). Reduced Collatz Dynamics is Periodical and the Period Equals 2 to the Power of the Count of x\/2. J. Math., submitted.","DOI":"10.36227\/techrxiv.11664501.v1"},{"key":"ref_5","unstructured":"Tomas Oliveira e Silva (2018, January 28). Computational Verification of the 3x+1 Conjecture. Available online: http:\/\/sweet.ua.pt\/tos\/3x+1.html."},{"key":"ref_6","unstructured":"Lagarias, J.C. (2010). Empirical Verification of the 3x+1 and Related Conjectures. The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ren, W., Li, S., Xiao, R., and Bi, W. (2018, January 8\u201312). Collatz Conjecture for 2^100000-1 is True\u2014Algorithms for Verifying Extremely Large Numbers. Proceedings of the 2018 IEEE SmartWorld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computing, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld\/SCALCOM\/UIC\/ATC\/CBDCom\/IOP\/SCI), Guangzhou, China. Available online: https:\/\/ieeexplore.ieee.org\/document\/8560077.","DOI":"10.1109\/SmartWorld.2018.00099"},{"key":"ref_8","unstructured":"Ren, W. (2019, May 01). Verifying Whether Extremely Large Integer Guarantees Collatz Conjecture (Can Return to 1 Finally), IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/fs3z-vc10."},{"key":"ref_9","unstructured":"Ren, W. (2019, May 01). Exploring the Ratio between the Count of x\/2 and the Count of (3*x+1)\/2 in original dynamics for extremely large starting integers asymptotically, IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/rxx6-8322."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ren, W. (2019). Collatz Dynamics is Partitioned by Residue Class Regularly. Inf. Comput., submitted.","DOI":"10.36227\/techrxiv.11742669"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5409","DOI":"10.1016\/j.tcs.2011.05.056","article-title":"The convergence classes of Collatz function","volume":"412","author":"Colussi","year":"2011","journal-title":"Theor. Comput. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0898-1221(92)90034-F","article-title":"3x + 1 Search Programs","volume":"24","author":"Leavens","year":"1992","journal-title":"Comput. Math. Appl."},{"key":"ref_13","first-page":"1281","article-title":"On the 3x + 100 problem","volume":"32","author":"Crandall","year":"1978","journal-title":"Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"237","DOI":"10.4064\/aa109-3-4","article-title":"Bounds for the 3x + 1 problem using difference inequalities","volume":"109","author":"Krasikov","year":"2003","journal-title":"Acta Arith."},{"key":"ref_15","unstructured":"Ren, W. (2019, May 01). Reduced Collatz Dynamics for Integers from 3 to 999999, IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/hq8c-x340."},{"key":"ref_16","unstructured":"Ren, W. (2019, May 01). Exploring the Inverse Mapping from a Dynamics to a Residue Class\u2014Inputting a Reduced Dynamics or Partial Dynamics and Outputting a Residue Class, IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/qmzw-gn71."},{"key":"ref_17","unstructured":"Ren, W. (2019, May 01). Collatz Automata and Compute Residue Class from Reduced Dynamics by Formula, IEEE Dataport. Available online: http:\/\/dx.doi.org\/10.21227\/7z84-ms87."}],"container-title":["Data"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2306-5729\/4\/2\/89\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:00:26Z","timestamp":1760187626000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2306-5729\/4\/2\/89"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,21]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["data4020089"],"URL":"https:\/\/doi.org\/10.3390\/data4020089","relation":{},"ISSN":["2306-5729"],"issn-type":[{"value":"2306-5729","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,21]]}}}