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Goldbach, Hurwitz, and the Infinitude of Primes: Weaving a Proof across the Centuries*

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References

  1. Alexanderson, G. L. (interviewer). 1985. “George Pólya,” in: Mathematical People: Profiles and Interviews, (Donald J. Albers and G. L. Alexanderson, eds.), Boston: Birkhäuser, pp. 246–253; see p. 250.

  2. Bell, E. T. 1937. Men of Mathematics, New York: Simon and Schuster. Includes lively semipopular accounts of the Bernoullis, Euler, and Fermat.

  3. Bellman, Richard. 1947. “A Note on Relatively Prime Sequences,” Bull. Am. Math. Soc. 53: 778–779.

    Google Scholar 

  4. Calinger, Ronald. 1996. “Leonhard Euler: The First St. Petersburg Years (1727–1741),” Historia Mathematica 23: 121–166.

  5. Dickson, L. E. 1952. History of the Theory of Numbers, Vol. 1: Divisibility and Primality, New York: Chelsea.

  6. Dunham, William. 1999. Euler: The Master of Us All, (Dolciani Mathematical Expositions, #22), Mathematical Association of America. Chapter 4, “Euler and Analytic Number Theory,” pp. 61–79, gives Euler’s infinitude-of-primes proof in full.

  7. Euler, Leonard. The Euler Archive, at http://www.eulerarchive.org. E72. 1744. Variae observationes circa series infinitas. E792. 1862. Tractatus de numerorum doctrina capita sedecim, quae supersunt. (Reprinted in Opera postuma 1, 1862, pp. 3–75.) Chapter 4: De numeris inter se primis et compositis, pp. 15–18.

  8. Euler, Leonhard. 1984. Elements of Algebra (translation by J. Hewlett of Vollständige Anleitung zur Algebra [Euler Archive E387], originally published: 5th ed., London: Longman, Orme, 1840), New York: Springer-Verlag.

  9. Euler, Leonard. 1988. Introduction to Analysis of the Infinite, Book 1 (translation by John D. Blanton of Introductio in analysin infinitorum, originally published 1748 [Euler Archive E101]), New York: Springer. See pp. 228–255, Chapter XV, “On Series Which Arise From Products”.

  10. Euler, Leonhard, and Goldbach, Christian. 1965. Briefwechsel 1729–1764 (A. P. Juškevič and E. Winter, eds.) (Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin), Berlin: Akademie Verlag.

  11. Freudenthal, Hans. 1972. “Hurwitz, Adolf,” Dictionary of Scientific Biography (Charles Coulston Gillispie, ed.), New York: Scribner’s, Vol. 4, pp. 570–573. See [H], below.

  12. Gale, David. 1991. “The Strange and Surprising Saga of the Somos Sequences” (“Mathematical Entertainments”), Math. Intelligencer 13(1): 40–42.

    Google Scholar 

  13. Hardy, G. H., and Wright, E. M. 1960. An Introduction to the Theory of Numbers, 4th ed., London: Oxford University Press.

  14. Hilbert, David. 1920. “Adolf Hurwitz” (Gedächtnisrede, May 15, 1920), reprinted in [Hu32], Vol. 1, pp. XIII–XX. Freudenthal laments the weakness of this eulogy, written late in Hilbert’s career: “because Hilbert wrote his biography, Hurwitz never got the one he deserved”.

  15. Hurwitz, Adolf. 1932–1933. Mathematische Werke, Basel: Birkhäuser.

  16. Hurwitz, A. 1972. Die Mathematischen Tagebücher und der übrige Handschriftliche Nachlass von Adolf Hurwitz (1859–1919): Katalog, (Alvin E. Jaeggli, ed.; Schriftenreihe der Bibliothek, 14), Zürich: Eidgenössische Technische Hochschule Zürich.

  17. Hurwitz, Adolf. 1993. Übungen zur Zahlentheorie, 1891–1918 (transcribed by Barbara Aquilino, prepared by Herbert Funk and Beat Glaus; Schriftenreihe der ETH-Bibliothek, 32), ETH ZÜrich University Archives.

  18. Jeltsch, R., and Mansour, M. (eds.). 1996. Stability Theory (Hurwitz Centenary Conference, Centro Stefano Franscini, Ascona, 1995), Basel: Birkhäuser.

  19. Juškevič, Adolf P., and Kopelevič, Judith Kh. 1994. Christian Goldbach: 1690–1764 (translated from Russian to German by Annerose and Walter Purkert, Vita mathematica, Band 8), Basel: Birkhäuser Verlag.

  20. Klein, Felix. 1928. Development of Mathematics in the 19th Century, Berlin: Springer-Verlag, (translated by M. Ackerman, Lie Groups: History, Frontiers and Applications, Vol. 9), Brookline, Mass.: Math Sci Press, 1979, p. 309.

  21. Mahoney, Michael S. 1972. “Goldbach, Christian,” Dictionary of Scientific Biography (Charles Coulston Gillispie, ed.), New York: Scribner’s, vol. 5, pp. 448–451.

  22. Mahoney, Michael Sean. 1994. The Mathematical Career of Pierre de Fermat 1601–1665 (2nd ed.), Princeton: Princeton University Press.

  23. Meissner, Ernst. 1919. “Gedächtnisrede auf Adolf Hurwitz” (November 21, 1919), reprinted in: [Hu32], Vol. 1, pp. XXI–XXIV.

  24. OEIS: The On-Line Encyclopedia of Integer Sequences, http://oeis.org.

  25. Parisse, D. 1997. The Tower of Hanoi and the Stern-Brocot Array, thesis, Munich.

  26. Pólya, George. 1969. “Some Mathematicians I Have Known,” Am. Math. Monthly 76: 746–753.

    Google Scholar 

  27. Pólya, G., and Szegő, G. 1925. Aufgaben und Lehrsätze aus der Analysis, Berlin: Springer (Dover reprint: New York, 1945), vol. 2, problem #94, pp. 133 and 342. English translation of 4th edition: Problems and Theorems in Analysis (trans. C. E. Billigheimer), New York: Springer, 1976, problem #94, pp. 130 and 322.

  28. Propp, James. “Somos Sequence Site,” http://www.math.wisc.edu/~propp/somos.html.

  29. Propp, James. http://www.math.harvard.edu/~propp/reach/shirt.html. The Somos sequence illustrated on a t-shirt.

  30. Reid, Constance. 1996. Hilbert, New York: Springer-Verlag, (originally Berlin: Springer-Verlag, 1970), esp. pp. 13–14.

  31. Ribenboim, Paulo. 1988. The Book of Prime Number Records, New York: Springer-Verlag.

  32. Ribenboim, Paulo. 1991. The Little Book of Big Primes, New York: Springer-Verlag.

  33. Ribenboim, Paulo. 1996. The New Book of Prime Number Records, New York: Springer-Verlag.

  34. Rowe, David E. 1986. “‘Jewish Mathematics’ at Göttingen in the Era of Felix Klein,” Isis 77: 422–449.

    Google Scholar 

  35. Rowe, David E. 2001. “Felix Klein as Wissenschaftspolitiker” in: Changing Images in Mathematics: From the French Revolution to the New Millennium (U. Bottazzini and A. D. Dalmedico, eds.; Studies in the History of Science, Technology and Medicine, Vol. 13), London: Routledge, pp. 69–91.

  36. Rowe, David E. 2007. “Felix Klein, Adolf Hurwitz, and the ‘Jewish Question’ in German Academia,” Math. Intelligencer 29(2): 18–30.

    Google Scholar 

  37. Sandifer, Ed. 2006. “Infinitely many primes,” How Euler Did It, MAA Online at http://www.maa.org/news/howeulerdidit.html, March, 2006.

  38. Somos, Michael. http://grail.cba.csuohio.edu/~somos/math.html.

  39. Somos, Michael, and Haas, Robert. 2003. “A Linked Pair of Sequences Implies the Primes Are Infinite,” Am. Math. Monthly 110: 539–540.

    Google Scholar 

  40. Struik, D. J. (ed.). 1969. A Source Book in Mathematics, 1200–1800, Cambridge, MA: Harvard University Press.

  41. Weil, André. 1984. Number Theory: An Approach through History, From Hammurabi to Legendre, Boston: Birkhäuser.

  42. Zelevinsky, Andrei. 2007. “What Is a Cluster Algebra?” Notices Am. Math. Soc. 54(11): 1494–1495. Somos sequence integrality explained by the Laurent phenomenon for cluster algebras.

    Google Scholar 

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Haas, R. Goldbach, Hurwitz, and the Infinitude of Primes: Weaving a Proof across the Centuries*. Math Intelligencer 36, 54–60 (2014). https://doi.org/10.1007/s00283-013-9402-8

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