References

Agrawal, Manindra, Neeraj Kayal, and Nitin Saxena. 2004. “PRIMES Is in P.” Annals of Mathematics 160 (2): 781–93.

Allouche, Jean-Paul, and Jeffrey Shallit. 2003. Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press.

Artin, Emil. 1927. Conjecture on Primitive Roots (Unpublished).

Bailey, David H., Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, and Victor H. Moll. 2007. Experimental Mathematics in Action. A K Peters.

Bailey, David H., Peter B. Borwein, and Simon Plouffe. 1997. “On the Rapid Computation of Various Polylogarithmic Constants.” Mathematics of Computation 66 (218): 903–13.

Barina, David. 2021. “Convergence Verification of the Collatz Problem.” Journal of Supercomputing 77: 2681–88.

Barnsley, Michael F. 1988. Fractals Everywhere. Academic Press.

Benford, Frank. 1938. “The Law of Anomalous Numbers.” Proceedings of the American Philosophical Society 78 (4): 551–72.

Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy. 1982. Winning Ways for Your Mathematical Plays. Vol. 2. Academic Press.

Bertrand, Joseph. 1887. “Solution d’un Problème.” Comptes Rendus de l’Académie Des Sciences 105: 369.

Borwein, Jonathan M., and Peter B. Borwein. 1987. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley.

Borwein, Jonathan M., and Keith Devlin. 2009. The Computer as Crucible: An Introduction to Experimental Mathematics. A K Peters.

Brin, Sergey, and Lawrence Page. 1998. “The Anatomy of a Large-Scale Hypertextual Web Search Engine.” Proceedings of the 7th International World Wide Web Conference, 107–17.

Cantor, Georg. 1883. “Über Unendliche, Lineare Punktmannigfaltigkeiten, V.” Mathematische Annalen 21: 545–91.

Carmichael, Robert Daniel. 1910. “Note on a New Number Theory Function.” Bulletin of the American Mathematical Society 16 (5): 232–38.

Champernowne, David Gawen. 1933. “The Construction of Decimals Normal in the Scale of Ten.” Journal of the London Mathematical Society 8: 254–60.

Conway, John Horton. 1987. “The Weird and Wonderful Chemistry of Audioactive Decay.” In Open Problems in Communication and Computation, edited by Thomas M. Cover and B. Gopinath. Springer.

Cook, Matthew. 2004. “Universality in Elementary Cellular Automata.” Complex Systems 15 (1): 1–40.

Copeland, Arthur Herbert, and Paul Erdős. 1946. “Note on Normal Numbers.” Bulletin of the American Mathematical Society 52: 857–60.

Cramér, Harald. 1936. “On the Order of Magnitude of the Difference Between Consecutive Prime Numbers.” Acta Arithmetica 2: 23–46.

Diffie, Whitfield, and Martin E. Hellman. 1976. “New Directions in Cryptography.” IEEE Transactions on Information Theory 22 (6): 644–54.

Eggenberger, Friedrich, and George Pólya. 1923. “Über Die Statistik Verketteter Vorgänge.” Zeitschrift für Angewandte Mathematik Und Mechanik 3 (4): 279–89.

Feigenbaum, Mitchell J. 1978. “Quantitative Universality for a Class of Nonlinear Transformations.” Journal of Statistical Physics 19 (1): 25–52.

Fomin, Sergey, and Andrei Zelevinsky. 2002. “The Laurent Phenomenon.” Advances in Applied Mathematics 28 (1): 119–44.

Gardner, Martin. 1967. “Mathematical Games: About the Dragon Curve and Other Problems.” Scientific American 216 (3): 124–25.

Gardner, Martin. 1970. “Mathematical Games: The Fantastic Combinations of John Conway’s New Solitaire Game ‘Life’.” Scientific American 223 (4): 120–23.

Gauss, Carl Friedrich. 1801. Disquisitiones Arithmeticae. Gerh. Fleischer jun.

Giuga, Giuseppe. 1950. “Su Una Presumibile Proprietà Caratteristica Dei Numeri Primi.” Istituto Lombardo Di Scienze e Lettere, Rendiconti, Classe Di Scienze Matematiche e Naturali 83: 511–28.

Goldbach, Christian. 1742. Letter to Leonhard Euler.

Golomb, Solomon Wolf. 1965. “A Self-Describing Sequence of Integers.” Mathematical Gazette 49: 173–74.

Gosper, Ralph William. 1972. Continued Fraction Arithmetic. Memo No. 239 (HAKMEM, Item 101B). Artificial Intelligence Laboratory, Massachusetts Institute of Technology.

Green, Ben, and Terence Tao. 2008. “The Primes Contain Arbitrarily Long Arithmetic Progressions.” Annals of Mathematics 167 (2): 481–547.

Guy, Richard K. 1988. “The Strong Law of Small Numbers.” American Mathematical Monthly 95 (8): 697–712.

Hardy, Godfrey Harold. 1940. A Mathematician’s Apology. Cambridge University Press.

Hardy, Godfrey Harold, and John Edensor Littlewood. 1923. “Some Problems of ‘Partitio Numerorum’; III: On the Expression of a Number as a Sum of Primes.” Acta Mathematica 44: 1–70.

Hardy, Godfrey Harold, and Edward Maitland Wright. 1979. An Introduction to the Theory of Numbers. 5th ed. Oxford University Press.

Haselgrove, Colin Brian. 1958. “A Disproof of a Conjecture of Pólya.” Mathematika 5: 141–45.

Hausdorff, Felix. 1918. “Dimension Und Äußeres Maß.” Mathematische Annalen 79: 157–79.

Heath, Thomas Little. 1897. The Works of Archimedes. Cambridge University Press.

Helfgott, Harald Andrés. 2013. The Ternary Goldbach Conjecture Is True.

Hénon, Michel. 1976. “A Two-Dimensional Mapping with a Strange Attractor.” Communications in Mathematical Physics 50 (1): 69–77.

Hurwitz, Adolf. 1891. “Über Die Angenäherte Darstellung Der Irrationalzahlen Durch Rationale Brüche.” Mathematische Annalen 39: 279–84.

Kakutani, Shizuo. 1944. “Two-Dimensional Brownian Motion and Harmonic Functions.” Proceedings of the Imperial Academy of Tokyo 20: 706–14.

Kanigel, Robert. 1991. The Man Who Knew Infinity: A Life of the Genius Ramanujan. Scribner.

Kaprekar, Dattatreya Ramchandra. 1949. “An Interesting Property of the Number 6174.” Scripta Mathematica 15: 244–45.

Kasiski, Friedrich Wilhelm. 1863. Die Geheimschriften Und Die Dechiffrierkunst. E. S. Mittler und Sohn.

Khinchin, Aleksandr Yakovlevich. 1964. Continued Fractions. University of Chicago Press.

Koch, Helge von. 1904. “Sur Une Courbe Continue Sans Tangente, Obtenue Par Une Construction Géométrique Élémentaire.” Arkiv för Matematik, Astronomi Och Fysik 1: 681–702.

Korselt, A. R. 1899. “Problème Chinois.” L’Intermédiaire Des Mathématiciens 6: 142–43.

Kummer, Ernst Eduard. 1852. “Über Die Ergänzungssätze Zu Den Allgemeinen Reciprocitätsgesetzen.” Journal für Die Reine Und Angewandte Mathematik 44: 93–146.

Kuzmin, Rodion. 1928. “Sur Un Problème de Gauss.” Atti Del Congresso Internazionale Dei Matematici 6: 83–89.

Lagarias, Jeffrey C. 1985. “The 3x+1 Problem and Its Generalizations.” American Mathematical Monthly 92 (1): 3–23.

Lagarias, Jeffrey C., Colin L. Mallows, and Allan R. Wilks. 2002. “Beyond the Descartes Circle Theorem.” American Mathematical Monthly 109 (4): 338–61.

Lagarias, Jeffrey C., Eric Rains, and Neil J. A. Sloane. 2002. “The EKG Sequence.” Experimental Mathematics 11 (3): 437–46.

Lagrange, Joseph-Louis. 1770. “Sur La Solution Des Problèmes Indéterminés Du Second Degré.” Mémoires de l’Académie Royale Des Sciences Et Belles-Lettres de Berlin 23: 165–310.

Lambert, Johann Heinrich. 1761. Mémoire Sur Quelques Propriétés Remarquables Des Quantités Transcendantes Circulaires Et Logarithmiques.

Lamé, Gabriel. 1844. “Note Sur La Limite Du Nombre Des Divisions Dans La Recherche Du Plus Grand Commun Diviseur Entre Deux Nombres Entiers.” Comptes Rendus de l’Académie Des Sciences de Paris 19: 867–70.

Langton, Chris G. 1990. “Computation at the Edge of Chaos: Phase Transitions and Emergent Computation.” Physica D: Nonlinear Phenomena 42 (1–3): 12–37.

Laplace, Pierre-Simon. 1812. Théorie Analytique Des Probabilités. Courcier.

Lawler, Gregory F. 1980. “A Self-Avoiding Random Walk.” Duke Mathematical Journal 47 (3): 655–93.

Lévy, Paul. 1929. “Sur Les Lois de Probabilité Dont dépendent Les Quotients Complets Et Incomplets d’une Fraction Continue.” Bulletin de La Société Mathématique de France 57: 178–94.

Li, Tien-Yien, and James A. Yorke. 1975. “Period Three Implies Chaos.” American Mathematical Monthly 82 (10): 985–92.

Lindemann, Carl Louis Ferdinand von. 1882. “Über Die Zahl π.” Mathematische Annalen 20: 213–25.

Lindenmayer, Aristid. 1968. “Mathematical Models for Cellular Interactions in Development I: Filaments with One-Sided Inputs.” Journal of Theoretical Biology 18 (3): 280–99.

Lorenz, Edward N. 1963. “Deterministic Nonperiodic Flow.” Journal of the Atmospheric Sciences 20 (2): 130–41.

Lucas, Édouard. 1878. “Théorie Des Fonctions Numériques Simplement périodiques.” American Journal of Mathematics 1 (2): 184–240.

Mandelbrot, Benoı̂t B. 1975. Les Objets Fractals: Forme, Hasard Et Dimension. Flammarion.

Mandelbrot, Benoı̂t B. 1980. “Fractal Aspects of the Iteration z → λz(1 − z) for Complex λ and z.” Annals of the New York Academy of Sciences 357: 249–59.

Markov, Andrei Andreevich. 1879. “Sur Les Formes Quadratiques Binàires Indéfinies.” Mathematische Annalen 15: 381–406.

Markus, Mario, and Benno Hess. 1990. “Lyapunov Exponents of the Logistic Map with Periodic Forcing.” Computers & Graphics 13 (4): 553–58.

Martin, Carla D. 2011. Keep It Real. Mathematical Association of America.

Menger, Karl. 1926. “Allgemeine Räume Und Cartesische Räume. Zweite Mitteilung: Über Umfassendste n-Dimensionale Mengen.” Koninklijke Akademie van Wetenschappen Te Amsterdam, Proceedings of the Section of Sciences 29: 1125–28.

Mertens, Franz. 1897. “Über Eine Zahlentheoretische Funktion.” Sitzungsberichte Der Kaiserlichen Akademie Der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse, Abteilung IIa 106: 761–830.

Milnor, John, and William Thurston. 1988. “On Iterated Maps of the Interval.” In Dynamical Systems, edited by J. C. Alexander, vol. 1342. Lecture Notes in Mathematics. Springer.

Minkowski, Hermann. 1904. “Zur Geometrie Der Zahlen.” Verhandlungen Des Dritten Internationalen Mathematiker-Kongresses in Heidelberg (Leipzig), 164–73.

Moivre, Abraham de. 1733. Approximatio Ad Summam Terminorum Binomii (a + b)ⁿ in Seriem Expansi.

O’Brien, Kevin, and Brian Beeler. 2025. “StorageReview Sets New Pi Record: 314 Trillion Digits on a Dell PowerEdge R7725.” https://www.storagereview.com/review/storagereview-sets-new-pi-record-314-trillion-digits-on-a-dell-poweredge-r7725.

Odlyzko, Andrew M., and Herman J. J. te Riele. 1985. “Disproof of the Mertens Conjecture.” Journal für Die Reine Und Angewandte Mathematik 357: 138–60.

Oliveira e Silva, Tomás, Siegfried Herzog, and Silvio Pardi. 2014. “Empirical Verification of the Even Goldbach Conjecture and Computation of Prime Gaps up to 4 × 10¹⁸.” Mathematics of Computation 83 (288): 2033–60.

Orr, William J. C. 1947. “Statistical Treatment of Polymer Solutions at Infinite Dilution.” Transactions of the Faraday Society 43: 12–27.

Pearson, Karl. 1905. “The Problem of the Random Walk.” Nature 72: 342.

Pólya, George. 1919. “Verschiedene Bemerkungen zur Zahlentheorie.” Jahresbericht Der Deutschen Mathematiker-Vereinigung 28: 31–40.

Pólya, George. 1921. “Über Eine Aufgabe Der Wahrscheinlichkeitsrechnung Betreffend Die Irrfahrt Im Straßennetz.” Mathematische Annalen 84: 149–60.

Prusinkiewicz, Przemysław, and Aristid Lindenmayer. 1990. The Algorithmic Beauty of Plants. Springer.

Rabinowitz, Stanley, and Stan Wagon. 1995. “A Spigot Algorithm for the Digits of Pi.” American Mathematical Monthly 102 (3): 195–203.

Ramanujan, Srinivasa. 1914. “Modular Equations and Approximations to π.” Quarterly Journal of Mathematics 45: 350–72.

Ramanujan, Srinivasa. 1919. “Some Properties of p(n), the Number of Partitions of n.” Proceedings of the Cambridge Philosophical Society 19: 207–10.

Richardson, Lewis Fry. 1961. “The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels.” General Systems Yearbook 6: 139–87.

Riemann, Bernhard. 1859. “Über Die Anzahl Der Primzahlen Unter Einer Gegebenen Größe.” Monatsberichte Der Königlich Preußischen Akademie Der Wissenschaften Zu Berlin, 671–80.

Rivest, Ronald L., Adi Shamir, and Leonard Adleman. 1978. “A Method for Obtaining Digital Signatures and Public-Key Cryptosystems.” Communications of the ACM 21 (2): 120–26.

Rowland, Eric S. 2008. “A Natural Prime-Generating Recurrence.” Journal of Integer Sequences 11 (2).

Sagan, Carl. 1985. Contact. Simon & Schuster.

Shamir, Adi. 1979. “How to Share a Secret.” Communications of the ACM 22 (11): 612–13.

Shanks, Daniel. 1971. “Class Number, a Theory of Factorization, and Genera.” In 1969 Number Theory Institute, vol. 20. Proceedings of Symposia in Pure Mathematics. American Mathematical Society.

Shannon, Claude E. 1948. “A Mathematical Theory of Communication.” Bell System Technical Journal 27 (3): 379–423.

Shannon, Claude E. 1949. “Communication Theory of Secrecy Systems.” Bell System Technical Journal 28 (4): 656–715.

Sharkovsky, Oleksandr Mykolayovych. 1964. “Co-Existence of Cycles of a Continuous Mapping of the Line into Itself.” Ukrains’kyi Matematychnyi Zhurnal 16 (1): 61–71.

Sierpiński, Wacław. 1915. “Sur Une Courbe Dont Tout Point Est Un Point de Ramification.” Comptes Rendus de l’Académie Des Sciences de Paris 160: 302–5.

Singmaster, David. 1971. “How Often Does an Integer Occur as a Binomial Coefficient?” American Mathematical Monthly 78 (4): 385–86.

Soddy, Frederick. 1936. “The Kiss Precise.” Nature 137: 1021.

Sós, Vera T. 1958. “On the Distribution Mod 1 of the Sequence nα.” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica 1: 127–34.

Stern, Moritz Abraham. 1858. “Über Eine Zahlentheoretische Funktion.” Journal für Die Reine Und Angewandte Mathematik 55: 193–220.

Tao, Terence. 2022. “Almost All Orbits of the Collatz Map Attain Almost Bounded Values.” Forum of Mathematics, Pi 10: e12.

Turing, Alan Mathison. 1936. “On Computable Numbers, with an Application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society 2 (42): 230–65.

Verhulst, Pierre-François. 1838. “Notice Sur La Loi Que La Population Suit Dans Son Accroissement.” Correspondance Mathématique Et Physique 10: 113–21.

Wallis, John. 1656. Arithmetica Infinitorum. Leon Lichfield.

Weierstrass, Karl. 1872. Über Continuirliche Functionen Eines Reellen Arguments, Die für Keinen Werth Des Letzteren Einen Bestimmten Differentialquotienten Besitzen.

Wiener, Norbert. 1923. “Differential-Space.” Journal of Mathematics and Physics 2: 131–74.

Wilson, David Bruce. 1996. “Generating Random Spanning Trees More Quickly Than the Cover Time.” Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 296–303.

Wolfram, Stephen. 1986. “Random Sequence Generation by Cellular Automata.” Advances in Applied Mathematics 7 (2): 123–69.

Wolfram, Stephen. 2002. A New Kind of Science. Wolfram Media.

Wolstenholme, Joseph. 1862. “On Certain Properties of Prime Numbers.” Quarterly Journal of Pure and Applied Mathematics 5: 35–39.

Zeckendorf, Edouard. 1972. “Représentation Des Nombres Naturels Par Une Somme de Nombres de Fibonacci Ou de Nombres de Lucas.” Bulletin de La Société Royale Des Sciences de Liège 41: 179–82.