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Doctoral research Bibliography

Bibliography

References and sources for my research on the Collatz conjecture and symbolic dynamics. Each entry carries tags —type, area, MSC, reading difficulty, access and importance to the thesis—; click any quality to see only that part of the bibliography, like the Cangur problems by topic.

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  1. [1]
    Althöfer, I. (2020). Lothar Collatz zwischen 1933 und 1950 – Eine Teilbiographie. 3-Hirn-Verlag. link ↗
    HistoricalHistory01A70AccessibleOpenImp. low
  2. [2]
    Applegate, D. & Lagarias, J. C. (2003). Lower bounds for the total stopping time of 3x+1 iterates. Math. Comp. 72(242), 1035–1049.
    ArticleNumber theoryComputation11B83HardPaywalledImp. medium
  3. [3]
    Bařina, D. (2025). Improved verification limit for the convergence of the Collatz conjecture. The Journal of Supercomputing. link ↗
    ArticleComputation11Y55MediumOpenImp. medium
  4. [4]
    Belaga, E. G. & Mignotte, M. (1998). Embedding the 3x+1 conjecture in a 3x+d context. Experimental Mathematics 7(2), 145–151. link ↗
    ArticleNumber theory11B83MediumPaywalledImp. medium
  5. [5]
    Belaga, E. G. & Mignotte, M. (2006). The Collatz problem and its generalizations: experimental data. Table 1: primitive cycles of (3n+d)-mappings. Prépublication IRMA, Strasbourg (HAL). link ↗
    PreprintNumber theoryComputation11B83MediumOpenImp. medium
  6. [6]
    Bennett, M. A. & de Weger, B. M. M. (1998). On the Diophantine equation |axn−byn|=1. Math. Comp. 67(221), 413–438.
    ArticleNumber theory11D61HardPaywalledImp. low
  7. [7]
    Bernstein, D. J. & Lagarias, J. C. (1996). The 3x+1 conjugacy map. Canadian Journal of Mathematics 48(6), 1154–1169.
    ArticleSymbolic dynamicsNumber theory11B83HardPaywalledImp. high
  8. [8]
    Böhm, C. & Sontacchi, G. (1978). 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. Atti Accad. Naz. Lincei (8) 64(3), 260–264.
    ArticleNumber theory11B83MediumLibraryImp. medium
  9. [9]
    Chamberland, M. (2003). An update on the 3x+1 problem. Butlletí de la Societat Catalana de Matemàtiques 18, 19–45.
    SurveyNumber theory11B83AccessibleOpenImp. medium
  10. [10]
    Collatz, L. & Sinogowitz, U. (1957). Spektren endlicher Grafen. Abh. Math. Sem. Univ. Hamburg 21, 63–77.
    HistoricalGraph theoryHistory05C50HardLibraryImp. low
  11. [11]
    Collatz, L. (1942). Einschließungssatz für die charakteristischen Zahlen von Matrizen. Math. Z. 48, 221–226.
    HistoricalHistory15A18HardLibraryImp. low
  12. [12]
    Collatz, L. (1986). On the motivation and origin of the (3n+1)-problem. En J. C. Lagarias (Ed.), The Ultimate Challenge: The 3x+1 Problem (pp. 241–247). AMS, 2010. link ↗
    HistoricalHistory01A60AccessiblePaywalledImp. medium
  13. [13]
    Coxeter, H. S. M. (1971). Cyclic sequences and frieze patterns. Vinculum 8, 4–7. [Reimpr. en The Ultimate Challenge, AMS 2010, 211–218.] link ↗
    HistoricalHistoryCombinatorics01A60AccessiblePaywalledImp. low
  14. [14]
    Crandall, R. E. (1978). On the “3x+1” problem. Math. Comp. 32(144), 1281–1292.
    ArticleNumber theory11B83HardOpenImp. high
  15. [15]
    Cvetković, D., Doob, M. & Sachs, H. (1995). Spectra of Graphs. Theory and Applications (3.ª ed.). Johann Ambrosius Barth.
    BookGraph theory05C50MediumLibraryImp. low
  16. [16]
    Eliahou, S. (1993). The 3x+1 problem: new lower bounds on nontrivial cycle lengths. Discrete Mathematics 118(1–3), 45–56.
    ArticleNumber theoryCombinatorics11B83HardPaywalledImp. high
  17. [17]
    Everett, C. J. (1977). Iteration of the number-theoretic function f(2n)=n, f(2n+1)=3n+2. Advances in Mathematics 25(1), 42–45.
    ArticleNumber theory11B83MediumPaywalledImp. medium
  18. [18]
    Gardner, M. (1972). Mathematical Games. Scientific American 226(6), 114–118.
    OutreachHistory00A08AccessiblePaywalledImp. low
  19. [19]
    Garner, L. E. (1985). On heights in the Collatz 3n+1 problem. Discrete Mathematics 55(1), 57–64.
    ArticleNumber theory11B83MediumPaywalledImp. low
  20. [20]
    Guy, R. K. (1983). Don't try to solve these problems! American Mathematical Monthly 90(1), 35–41. link ↗
    OutreachHistory00A08AccessiblePaywalledImp. low
  21. [21]
    Hayes, B. (1984). Computer recreations: The ups and downs of hailstone numbers. Scientific American 250(1), 10–16.
    OutreachHistoryComputation00A08AccessiblePaywalledImp. low
  22. [22]
    Holden, D. (2018). Results on the 3x+1 and 3x+d conjectures. Fibonacci Quarterly (preprint, CSU Fresno).
    ArticleNumber theory11B83MediumLibraryImp. medium
  23. [23]
    Karras, I. & de Weger, B. M. M. (2026). Modular Collatz graphs (Determinants of Modular Collatz Graphs and Variants). Preprint, arXiv:2601.15463. link ↗
    PreprintGraph theorySymbolic dynamics11B83HardOpenImp. high
  24. [24]
    Krasikov, I. & Lagarias, J. C. (2003). Bounds for the 3x+1 problem using difference inequalities. Acta Arithmetica 109(3), 237–258.
    ArticleNumber theoryAnalysis & probability11B83HardPaywalledImp. medium
  25. [25]
    Laarhoven, T. & de Weger, B. (2013). The Collatz conjecture and De Bruijn graphs. Indagationes Mathematicae (N.S.) 24(4), 971–983. link ↗
    ArticleGraph theorySymbolic dynamics11B83MediumPaywalledImp. high
  26. [26]
    Lagarias, J. C. (1985). The 3x+1 problem and its generalizations. American Mathematical Monthly 92(1), 3–23. link ↗
    SurveyNumber theory11B83AccessibleOpenImp. high
  27. [27]
    Lagarias, J. C. (1990). The set of rational cycles for the 3x+1 problem. Acta Arithmetica 56(1), 33–53.
    ArticleNumber theory11B83HardPaywalledImp. medium
  28. [28]
    Lagarias, J. C. (2003). The 3x+1 problem: an annotated bibliography (1963–1999). arXiv:math/0309224. link ↗
    SurveyNumber theoryHistory11B83AccessibleOpenImp. medium
  29. [29]
    Lagarias, J. C. (2010). The 3x+1 problem: an overview. En The Ultimate Challenge, AMS, 3–29. (arXiv:2111.02635.) link ↗
    SurveyNumber theory11B83AccessibleOpenImp. high
  30. [30]
    Lagarias, J. C. (Ed.) (2010). The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society. link ↗
    BookNumber theory11B83MediumPaywalledImp. high
  31. [31]
    Lang, W. (2014). On Collatz words, sequences and trees. Journal of Integer Sequences 17, Art. 14.11.7. link ↗
    ArticleSymbolic dynamicsCombinatorics11B83MediumOpenImp. high
  32. [32]
    Laurent, M., Mignotte, M. & Nesterenko, Y. (1995). Formes linéaires en deux logarithmes et déterminants d'interpolation. Journal of Number Theory 55(2), 285–321.
    ArticleNumber theory11J86HardPaywalledImp. low
  33. [33]
    Lind, D. & Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding. Cambridge University Press. link ↗
    BookSymbolic dynamicsDynamical systems37B10MediumPaywalledImp. high
  34. [34]
    Mathematics Genealogy Project, id 20676 (Lothar Collatz). link ↗
    ResourceHistory01A60AccessibleOpenImp. low
  35. [35]
    Matthews, K. R. & Watts, A. M. (1984). A generalization of Hasse's generalization of the Syracuse algorithm. Acta Arithmetica 43(2), 167–175.
    ArticleNumber theory11B83HardPaywalledImp. medium
  36. [36]
    Monks, K. G. (2006). The sufficiency of arithmetic progressions for the 3x+1 conjecture. Proceedings of the AMS 134(10), 2861–2872. link ↗
    ArticleNumber theory11B83MediumOpenImp. medium
  37. [37]
    Moreau, C. (1872). Sur les permutations circulaires distinctes. Nouvelles Annales de Mathématiques (2) 11, 309–314. link ↗
    ArticleCombinatorics05A05AccessibleOpenImp. high
  38. [38]
    O'Connor, J. J. & Robertson, E. F. (2006). Lothar Collatz. MacTutor, Univ. of St Andrews. link ↗
    ResourceHistory01A70AccessibleOpenImp. low
  39. [39]
    Oliveira e Silva, T. (1999). Maximum excursion and stopping time record-holders for the 3x+1 problem: computational results. Math. Comp. 68(225), 371–384. link ↗
    ArticleComputation11Y55MediumOpenImp. low
  40. [40]
    Reutenauer, C. (1993). Free Lie Algebras. Oxford University Press.
    BookAlgebra & wordsCombinatorics17B01HardLibraryImp. medium
  41. [41]
    Sander, J. W. (1990). On the (3N+1)-conjecture. Acta Arithmetica 55(3), 241–248.
    ArticleNumber theory11B83MediumPaywalledImp. low
  42. [42]
    Segal, S. L. (2003). Mathematicians under the Nazis. Princeton University Press.
    HistoricalHistory01A60AccessiblePaywalledImp. low
  43. [43]
    Simons, J. L. & de Weger, B. (2005). Theoretical and computational bounds for m-cycles of the 3n+1 problem. Acta Arithmetica 117(1), 51–70. link ↗
    ArticleNumber theory11B83HardPaywalledImp. high
  44. [44]
    Simons, J. L. (2005). On the nonexistence of 2-cycles for the 3x+1 problem. Math. Comp. 74, 1565–1572. link ↗
    ArticleNumber theory11B83HardOpenImp. medium
  45. [45]
    Simons, J. L. (2008). On the (non-)existence of m-cycles for generalized Syracuse sequences. Acta Arithmetica 131(3), 217–254. link ↗
    ArticleNumber theory11B83HardPaywalledImp. medium
  46. [46]
    Steiner, R. P. (1978). A theorem on the Syracuse problem. Proc. 7th Manitoba Conf. on Numerical Mathematics and Computing, 553–559.
    ArticleNumber theory11B83MediumLibraryImp. medium
  47. [47]
    Tao, T. (2022). Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi 10, e12. link ↗
    ArticleAnalysis & probabilityNumber theory11B83HardOpenImp. high
  48. [48]
    Terras, R. (1976). A stopping time problem on the positive integers. Acta Arithmetica 30(3), 241–252.
    ArticleNumber theoryAnalysis & probability11B83MediumPaywalledImp. high
  49. [49]
    Thwaites, B. (1985). My conjecture. Bull. Inst. Math. Appl. 21, 35–41.
    HistoricalHistory01A60MediumLibraryImp. low
  50. [50]
    Tremblay, R. (2020). Generalized 3x+1 mappings: counting cycles. arXiv:2008.11103. link ↗
    PreprintCombinatoricsNumber theory11B83MediumOpenImp. medium
  51. [51]
    Wirsching, G. J. (1996). On the combinatorial structure of 3N+1 predecessor sets. Discrete Mathematics 148(1–3), 265–286.
    ArticleCombinatoricsNumber theory11B83HardPaywalledImp. medium
  52. [52]
    Wirsching, G. J. (1998). The Dynamical System Generated by the 3n+1 Function. Springer, Lecture Notes in Mathematics 1681. link ↗
    BookDynamical systemsNumber theory11B83HardPaywalledImp. high