Collatz conjecture (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Collatz conjecture" in English language version.

refsWebsite

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19doi.org

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4ams.org

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3arxiv.org

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3worldcat.org

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3semanticscholar.org

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3harvard.edu

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2jstor.org

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2archive.org

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2books.google.com

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2zbmath.org

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1st-andrews.ac.uk

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1ericr.nl

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1springer.com

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1web.archive.org

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1sfu.ca

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1icm.edu.pl

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1quantamagazine.org

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1impan.pl

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1uwaterloo.ca

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1eudml.org

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1emis.de

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1unict.it

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1claremont.edu

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1uchicago.edu

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ams.org

mathscinet.ams.org

Terras, Riho (1976). "A stopping time problem on the positive integers" (PDF). Acta Arithmetica. 30 (3): 241–252. doi:10.4064/aa-30-3-241-252. MR 0568274.
Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
Steiner, R. P. (1977). "A theorem on the syracuse problem". Proceedings of the 7th Manitoba Conference on Numerical Mathematics. pp. 553–9. MR 0535032.
Simons, John L. (2005). "On the nonexistence of 2-cycles for the 3x + 1 problem". Math. Comp. 74: 1565–72. Bibcode:2005MaCom..74.1565S. doi:10.1090/s0025-5718-04-01728-4. MR 2137019.

archive.org

Pickover, Clifford A. (2001). Wonders of Numbers. Oxford: Oxford University Press. pp. 116–118. ISBN 0-19-513342-0.
Hofstadter, Douglas R. (1979). Gödel, Escher, Bach. New York: Basic Books. pp. 400–2. ISBN 0-465-02685-0.

arxiv.org

Tao, Terence (2022). "Almost all orbits of the Collatz map attain almost bounded values". Forum of Mathematics, Pi. 10: e12. arXiv:1909.03562. doi:10.1017/fmp.2022.8. ISSN 2050-5086.
Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
"Hardness of busy beaver value BB(15)".

books.google.com

Guy, Richard K. (2004). ""E16: The 3x+1 problem"". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. pp. 330–6. ISBN 0-387-20860-7. Zbl 1058.11001.
Kurtz, Stuart A.; Simon, Janos (2007). "The undecidability of the generalized Collatz problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 978-3-540-72503-9. As PDF

claremont.edu

scholarship.claremont.edu

Clay, Oliver Keatinge. "The Long Search for Collatz Counterexamples". p. 208. Retrieved 26 July 2024.

doi.org

Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". The American Mathematical Monthly. 92 (1): 3–23. doi:10.1080/00029890.1985.11971528. JSTOR 2322189.
Tao, Terence (2022). "Almost all orbits of the Collatz map attain almost bounded values". Forum of Mathematics, Pi. 10: e12. arXiv:1909.03562. doi:10.1017/fmp.2022.8. ISSN 2050-5086.
Leavens, Gary T.; Vermeulen, Mike (December 1992). "3x + 1 search programs". Computers & Mathematics with Applications. 24 (11): 79–99. doi:10.1016/0898-1221(92)90034-F.
Barina, David (2025). "Improved verification limit for the convergence of the Collatz conjecture" (PDF). The Journal of Supercomputing. 81 (810): 1–14. doi:10.1007/s11227-025-07337-0. S2CID 220294340.
Garner, Lynn E. (1981). "On the Collatz 3n + 1 algorithm". Proceedings of the American Mathematical Society. 82 (1): 19–22. doi:10.1090/S0002-9939-1981-0603593-2. JSTOR 2044308.
Eliahou, Shalom (1993). "The 3x + 1 problem: new lower bounds on nontrivial cycle lengths". Discrete Mathematics. 118 (1): 45–56. doi:10.1016/0012-365X(93)90052-U.
Simons, J.; de Weger, B. (2005). "Theoretical and computational bounds for m-cycles of the 3n + 1 problem" (PDF). Acta Arithmetica. 117 (1): 51–70. Bibcode:2005AcAri.117...51S. doi:10.4064/aa117-1-3. Archived from the original on 2022-03-18. Retrieved 2023-03-28.{{cite journal}}: CS1 maint: bot: original URL status unknown (link)
Terras, Riho (1976). "A stopping time problem on the positive integers" (PDF). Acta Arithmetica. 30 (3): 241–252. doi:10.4064/aa-30-3-241-252. MR 0568274.
Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
Simons, John L. (2005). "On the nonexistence of 2-cycles for the 3x + 1 problem". Math. Comp. 74: 1565–72. Bibcode:2005MaCom..74.1565S. doi:10.1090/s0025-5718-04-01728-4. MR 2137019.
Colussi, Livio (9 September 2011). "The convergence classes of Collatz function". Theoretical Computer Science. 412 (39): 5409–5419. doi:10.1016/j.tcs.2011.05.056.
Hew, Patrick Chisan (7 March 2016). "Working in binary protects the repetends of 1/3^h: Comment on Colussi's 'The convergence classes of Collatz function'". Theoretical Computer Science. 618: 135–141. doi:10.1016/j.tcs.2015.12.033.
Lagarias, Jeffrey (1990). "The set of rational cycles for the 3x+1 problem". Acta Arithmetica. 56 (1): 33–53. doi:10.4064/aa-56-1-33-53. ISSN 0065-1036.
Belaga, Edward G.; Mignotte, Maurice (1998). "Embedding the 3x+1 Conjecture in a 3x+d Context". Experimental Mathematics. 7 (2): 145–151. doi:10.1080/10586458.1998.10504364. S2CID 17925995.
Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996). "The 3x + 1 conjugacy map". Canadian Journal of Mathematics. 48 (6): 1154–1169. doi:10.4153/CJM-1996-060-x. ISSN 0008-414X.
Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-problem and holomorphic dynamics". Experimental Mathematics. 8 (3): 241–252. doi:10.1080/10586458.1999.10504402.
Kurtz, Stuart A.; Simon, Janos (2007). "The undecidability of the generalized Collatz problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 978-3-540-72503-9. As PDF
Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability. 1 (1): 19–56. doi:10.3233/COM-150032.
Michel, Pascal (1993). "Busy beaver competition and Collatz-like problems". Archive for Mathematical Logic. 32 (5): 351–367. doi:10.1007/BF01409968.

emis.de

Belaga, Edward G.; Mignotte, Maurice (1998). "Embedding the 3x+1 Conjecture in a 3x+d Context". Experimental Mathematics. 7 (2): 145–151. doi:10.1080/10586458.1998.10504364. S2CID 17925995.

ericr.nl

Roosendaal, Eric. "3x+1 delay records". Retrieved 14 March 2020. (Note: "Delay records" are total stopping time records.)

eudml.org

Lagarias, Jeffrey (1990). "The set of rational cycles for the 3x+1 problem". Acta Arithmetica. 56 (1): 33–53. doi:10.4064/aa-56-1-33-53. ISSN 0065-1036.

harvard.edu

ui.adsabs.harvard.edu

Simons, J.; de Weger, B. (2005). "Theoretical and computational bounds for m-cycles of the 3n + 1 problem" (PDF). Acta Arithmetica. 117 (1): 51–70. Bibcode:2005AcAri.117...51S. doi:10.4064/aa117-1-3. Archived from the original on 2022-03-18. Retrieved 2023-03-28.{{cite journal}}: CS1 maint: bot: original URL status unknown (link)
Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
Simons, John L. (2005). "On the nonexistence of 2-cycles for the 3x + 1 problem". Math. Comp. 74: 1565–72. Bibcode:2005MaCom..74.1565S. doi:10.1090/s0025-5718-04-01728-4. MR 2137019.

icm.edu.pl

matwbn.icm.edu.pl

Terras, Riho (1976). "A stopping time problem on the positive integers" (PDF). Acta Arithmetica. 30 (3): 241–252. doi:10.4064/aa-30-3-241-252. MR 0568274.

impan.pl

Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.

jstor.org

Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". The American Mathematical Monthly. 92 (1): 3–23. doi:10.1080/00029890.1985.11971528. JSTOR 2322189.
Garner, Lynn E. (1981). "On the Collatz 3n + 1 algorithm". Proceedings of the American Mathematical Society. 82 (1): 19–22. doi:10.1090/S0002-9939-1981-0603593-2. JSTOR 2044308.

quantamagazine.org

Hartnett, Kevin (December 11, 2019). "Mathematician Proves Huge Result on 'Dangerous' Problem". Quanta Magazine.

semanticscholar.org

api.semanticscholar.org

Barina, David (2025). "Improved verification limit for the convergence of the Collatz conjecture" (PDF). The Journal of Supercomputing. 81 (810): 1–14. doi:10.1007/s11227-025-07337-0. S2CID 220294340.
Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
Belaga, Edward G.; Mignotte, Maurice (1998). "Embedding the 3x+1 Conjecture in a 3x+d Context". Experimental Mathematics. 7 (2): 145–151. doi:10.1080/10586458.1998.10504364. S2CID 17925995.

sfu.ca

cecm.sfu.ca

Lagarias (1985),^[2] section "A heuristic argument".

springer.com

link.springer.com

Barina, David (2025). "Improved verification limit for the convergence of the Collatz conjecture" (PDF). The Journal of Supercomputing. 81 (810): 1–14. doi:10.1007/s11227-025-07337-0. S2CID 220294340.

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

O'Connor, John J.; Robertson, Edmund F., "Lothar Collatz", MacTutor History of Mathematics Archive, University of St Andrews

uchicago.edu

cs.uchicago.edu

Kurtz, Stuart A.; Simon, Janos (2007). "The undecidability of the generalized Collatz problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 978-3-540-72503-9. As PDF

unict.it

dmi.unict.it

Scollo, Giuseppe (2007). "Looking for class records in the 3x + 1 problem by means of the COMETA grid infrastructure" (PDF). Grid Open Days at the University of Palermo.

uwaterloo.ca

cs.uwaterloo.ca

Hercher, C. (2023). "There are no Collatz m-cycles with m <= 91" (PDF). Journal of Integer Sequences. 26 (3): Article 23.3.5.

web.archive.org

Simons, J.; de Weger, B. (2005). "Theoretical and computational bounds for m-cycles of the 3n + 1 problem" (PDF). Acta Arithmetica. 117 (1): 51–70. Bibcode:2005AcAri.117...51S. doi:10.4064/aa117-1-3. Archived from the original on 2022-03-18. Retrieved 2023-03-28.{{cite journal}}: CS1 maint: bot: original URL status unknown (link)

worldcat.org

search.worldcat.org

Tao, Terence (2022). "Almost all orbits of the Collatz map attain almost bounded values". Forum of Mathematics, Pi. 10: e12. arXiv:1909.03562. doi:10.1017/fmp.2022.8. ISSN 2050-5086.
Lagarias, Jeffrey (1990). "The set of rational cycles for the 3x+1 problem". Acta Arithmetica. 56 (1): 33–53. doi:10.4064/aa-56-1-33-53. ISSN 0065-1036.
Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996). "The 3x + 1 conjugacy map". Canadian Journal of Mathematics. 48 (6): 1154–1169. doi:10.4153/CJM-1996-060-x. ISSN 0008-414X.

zbmath.org

Guy, Richard K. (2004). ""E16: The 3x+1 problem"". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. pp. 330–6. ISBN 0-387-20860-7. Zbl 1058.11001.
Lagarias, Jeffrey C., ed. (2010). The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society. ISBN 978-0-8218-4940-8. Zbl 1253.11003.