{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:45:04Z","timestamp":1774626304665,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,5]],"date-time":"2021-02-05T00:00:00Z","timestamp":1612483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to the visualization of behaviors of the Collatz iteration function and the analysis of the results.<\/jats:p>","DOI":"10.3390\/math9040314","type":"journal-article","created":{"date-parts":[[2021,2,5]],"date-time":"2021-02-05T03:34:02Z","timestamp":1612496042000},"page":"314","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Clustering Perspective of the Collatz Conjecture"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"Jos\u00e9 A. Tenreiro","family":"Machado","sequence":"first","affiliation":[{"name":"Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8262-1369","authenticated-orcid":false,"given":"Alexandra","family":"Galhano","sequence":"additional","affiliation":[{"name":"Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2266-2075","authenticated-orcid":false,"given":"Daniel","family":"Cao Labora","sequence":"additional","affiliation":[{"name":"Department of Statistics, Mathematical Analysis and Optimization, Faculty of Mathematics, Institute of Mathematics (IMAT), Universidade de Santiago de Compostela (USC), R\u00faa Lope G\u00f3mez de Marzoa s\/n, 15782 Santiago de Compostela, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lagarias, J. (2010). The Ultimate Challenge: The 3x + 1 Problem, American Mathematical Society.","DOI":"10.1090\/mbk\/078"},{"key":"ref_2","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","year":"2003","journal-title":"Acta Arith."},{"key":"ref_3","unstructured":"Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0012-365X(93)90052-U","article-title":"The 3x + 1 problem: New lower bounds on nontrivial cycle lengths","volume":"118","author":"Eliahou","year":"1993","journal-title":"Discret. Math."},{"key":"ref_5","first-page":"260","article-title":"On the existence of cycles of given length in integer sequences like xn+1 = xn\/2 if xn even, and xn+1 = 3xn + 1 otherwise","volume":"64","author":"Sontacchi","year":"1978","journal-title":"Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat."},{"key":"ref_6","unstructured":"Andrei, S., Kudlek, M., and Niculescu, R.S. (1999). Chains in Collatz\u2019s Tree, Department of Informatics, Universit\u00e4t Hamburg. Technical Report."},{"key":"ref_7","first-page":"43","article-title":"The 3x + 1 Problem and Directed Graphs","volume":"40","author":"Andaloro","year":"2002","journal-title":"Fibonacci Q."},{"key":"ref_8","first-page":"1","article-title":"The Collatz Problem and Analogues","volume":"11","author":"Snapp","year":"2008","journal-title":"J. Integer Seq."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1016\/j.indag.2013.03.003","article-title":"The Collatz conjecture and De Bruijn graphs","volume":"24","author":"Laarhoven","year":"2013","journal-title":"Indag. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Emmert-Streib, F. (2013). Structural Properties and Complexity of a New Network Class: Collatz Step Graphs. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0056461"},{"key":"ref_11","unstructured":"Sultanow, E., Koch, C., and Cox, S. (2019). Collatz Sequences in the Light of Graph Theory, Universit\u00e4t Potsdam."},{"key":"ref_12","unstructured":"Ebert, H. (2020). A Graph Theoretical Approach to the Collatz Problem. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02289565","article-title":"Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis","volume":"29","author":"Kruskal","year":"1964","journal-title":"Psychometrika"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kruskal, J.B., and Wish, M. (1978). Multidimensional Scaling, Sage Publications.","DOI":"10.4135\/9781412985130"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/T-C.1969.222678","article-title":"A nonlinear mapping for data structure analysis","volume":"C-18","author":"Sammon","year":"1969","journal-title":"IEEE Trans. Comput."},{"key":"ref_16","unstructured":"Hartigan, J.A. (1975). Clustering Algorithms, John Wiley & Sons, Inc."},{"key":"ref_17","unstructured":"Borg, I., and Groenen, P.J. (2005). Modern Multidimensional Scaling-Theory and Applications, Springer."},{"key":"ref_18","first-page":"1","article-title":"Multidimensional scaling using majorization: Smacof in R","volume":"31","author":"Mair","year":"2009","journal-title":"J. Stat. Softw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/BF02289630","article-title":"The analysis of proximities: Multidimensional scaling with an unknown distance function","volume":"27","author":"Shepard","year":"1962","journal-title":"Psychometrika"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s00357-008-9004-x","article-title":"Solving non-uniqueness in agglomerative hierarchical clustering using multidendrograms","volume":"25","year":"2008","journal-title":"J. Classif."},{"key":"ref_21","first-page":"47","article-title":"A Survey on Multidimensional Scaling","volume":"51","author":"Saeed","year":"2018","journal-title":"ACM Comput. Surv. (CSUR)"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105128","DOI":"10.1016\/j.cnsns.2019.105128","article-title":"Multidimensional scaling and visualization of patterns in prime numbers","volume":"83","author":"Machado","year":"2020","journal-title":"Commun. Nonlinea. Sci. Numer. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Machado, J.A.T. (2020). An Evolutionary Perspective of Virus Propagation. Mathematics, 8.","DOI":"10.3390\/math8050779"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"104883","DOI":"10.1016\/j.cnsns.2019.104883","article-title":"A computational perspective of the periodic table of elements","volume":"78","author":"Machado","year":"2019","journal-title":"Commun. Nonlinea. Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.jare.2020.01.004","article-title":"Multidimensional scaling locus of memristor and fractional order elements","volume":"25","author":"Machado","year":"2020","journal-title":"J. Adv. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1007\/s11071-020-05771-8","article-title":"Computational analysis of the SARS-CoV-2 and other viruses based on the Kolmogorov\u2019s complexity and Shannon\u2019s information theories","volume":"101","author":"Machado","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.3390\/e17041775","article-title":"Multidimensional Scaling Visualization Using Parametric Similarity Indices","volume":"17","author":"Lopes","year":"2015","journal-title":"Entropy"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Aggarwal, C.C., Hinneburg, A., and Keim, D.A. (2001). On the Surprising Behavior of Distance Metrics in Gigh Dimensional Space, Springer.","DOI":"10.1007\/3-540-44503-X_27"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Sokal, R.R., and Rohlf, F.J. (1962). The comparison of dendrograms by objective methods. Taxon, 33\u201340.","DOI":"10.2307\/1217208"},{"key":"ref_30","unstructured":"Felsenstein, J. (1993). PHYLIP (Phylogeny Inference Package), Version 3.5 c, University of Washington."},{"key":"ref_31","unstructured":"Tuimala, J. (2006). A Primer to Phylogenetic Analysis Using the PHYLIP Package, CSC\u2014Scientific Computing Ltd."},{"key":"ref_32","first-page":"300","article-title":"Comprehensive Survey on Distance\/Similarity Measures between Probability Density Functions","volume":"1","author":"Cha","year":"2007","journal-title":"Int. J. Math. Models Methods Appl. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Deza, M.M., and Deza, E. (2009). Encyclopedia of Distances, Springer-Verlag.","DOI":"10.1007\/978-3-642-00234-2"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1002\/j.1538-7305.1950.tb00463.x","article-title":"Error Detecting and Error Correcting Codes","volume":"29","author":"Hamming","year":"1950","journal-title":"Bell Syst. Tech. J."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/9\/4\/314\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:20:30Z","timestamp":1760160030000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/9\/4\/314"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,5]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["math9040314"],"URL":"https:\/\/doi.org\/10.3390\/math9040314","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,5]]}}}