Abstract
In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews G.E. (1983) Euler’s pentagonal number theorem. Mathematics Magazine 56(5): 279–284
Andrews, G.E. 1998. The theory of partitions. Cambridge University Press.
Bump, D. 1998. Automorphic forms and representations, vol. 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press.
Cajori F. (1918) Origin of the name “mathematical induction”. American Mathematical Monthly 25(5): 197–201
Cajori F. (1929) A History of Mathematical Notations, vol. II. The Open Court Publishing Company, Chicago
Calinger R. (1996) Leonhard Euler: the first St. Petersburg years (1727–1741). Historia Mathematica 23(2): 121–166
Cantor, M. 1901. Vorlesungen über Geschichte der Mathematik. Dritter Band. Vom Jahre 1668 biz zum Jahre 1758. Leipzig: B. G. Teubner, Zweite Auflage. Reprinted by Johnson Reprint Corporation, New York, 1965.
Cartan, H. 1995. Elementary theory of analytic functions of one or several complex variables. Mineola, NY: Dover Publications. Translated from the French.
Chandrasekharan K. (1985) Elliptic functions, Die Grundlehren der mathematischen Wissenschaften, vol. 281. Springer-Verlag, Berlin
Chrystal, G. 1964. Algebra: an elementary text-book, Part II. New York: Chelsea Publishing Company, seventh edition. Reprinted by AMS Chelsea Publishing, 1999.
Cramer, G., editor. 1744. Jacobi Bernoulli, Basileensis, Opera. Tomus Secundus. Sumptibus Haeredum Cramer & Fratrum Philibert=Héritiers Cramer et frères Philibert, Genevae. With the editorial assistance of Nicolaus I Bernoulli.
Dickson, L.E. 1919a. History of the theory of numbers, vol. I. Washington, D.C.: Carnegie Institute. Publication Number 256.
Dickson, L.E. 1919b. History of the theory of numbers, vol. II. Washington, D.C.: Carnegie Institute. Publication Number 256.
Eneström G. (1906) Der Briefwechsel zwischen Leonhard Euler und Daniel Bernoulli. Bibl. Math., 3. Folge 7: 126–156
Eneström G. (1908) Jakob Bernoulli und die Jacobischen Thetafunktionen. Bibl. Math., 3. Folge 9: 206–210
Eneström, G. 1910/1913. Verzeichnis der Schriften Leonhard Eulers. B. G. Teubner, Leipzig. Jahresbericht der Deutschen Mathematiker-Vereinigung, Ergänzungsband IV, Lief. 1, 2.
Euler, L. 1738. Methodus generalis summandi progressiones. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 6: 68–97. E25, Opera omnia I.14, pp. 42–72.
Euler, L. 1740. De summis serierum reciprocarum. Commentarii Academiae Scientiarum Imperialis Petropolitanae 7: 123–134. E41, Opera omnia I.14, pp. 73–86.
Euler, L. 1744. Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 9: 160–188. E72, Opera omnia I.14, pp. 217–244.
Euler, L. 1748. Introductio in analysin infinitorum. Tomus primus. Apud Marcum-Michaelem Bousquet & Socios=Marc-Michel Bousquet & Compagnie, Lausannae. E101, Opera omnia I.8.
Euler, L. 1751a. Découverte d’une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs. Bibliothèque Impartiale, 3(1): 10–31. E175, Opera omnia I.2, pp. 241–253.
Euler, L. 1751b. Observationes analyticae variae de combinationibus. Commentarii Academiae Scientiarum Imperialis Petropolitanae 13: 64–93. E158, Opera omnia I.2, pp. 163–193.
Euler, L. 1753a. Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 3: 86–108. E190, Opera omnia I.14, pp. 516–541.
Euler, L. 1753b. De partitione numerorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 3: 125–169. E191, Opera omnia I.2, pp. 254–294.
Euler, L. 1760a. Demonstratio theorematis circa ordinem in summis divisorum observatum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 5: 75–83. E244, Opera omnia I.2, pp. 390–398.
Euler, L. 1760b. Observatio de summis divisorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 5: 59–74. E243, Opera omnia I.2, pp. 373–389.
Euler, L. 1768. Lettres à une princesse d’Allemagne sur divers sujets de physique & philosophie, Tome second. Academie Impériale des Sciences, Saint Petersbourg. E344, Opera omnia III.12.
Euler, L. 1770. Vollständige Anleitung zur Algebra. Erster Theil. Kays. Acad. der Wissenschaften, St. Petersburg. E387, Opera omnia I.1.
Euler, L. 1783a. De mirabilibus proprietatibus numerorum pentagonalium. Acta Academiae Scientiarum Imperialis Petropolitanae 4: 56–75. E542, Opera omnia I.3, pp. 480–496.
Euler, L. 1783b. Evolutio producti infiniti (1 − x)(1 − xx)(1 − x 3)(1 − x 4)(1 − x 5)(1 − x 6) etc. in seriem simplicem. Acta Academiae Scientiarum Imperialis Petropolitanae 4: 47–55. E541, Opera omnia I.3, pp. 472–479.
Euler, L. 1784. De plurimis quantitatibus transcendentibus, quas nullo modo per formulas integrales exprimere licet. Acta Academiae Scientiarum Imperialis Petropolitanae 4: 31–37. E565, Opera omnia I.15, pp. 522–527.
Euler, L. 1785. Considerationes super theoremate Fermatiano de resolutione numerorum in numeros polygonales. Opuscula Analytica 2: 3–15. E586, Opera omnia I.4, pp. 125–135.
Fellmann, E.A. 1995. Leonhard Euler. Rowohlt Taschenbuch, Reinbek bei Hamburg. Rowohlts Monographien, Band 387.
Fellmann, E.A., Mikhajlov, G.K. (eds) (1998) Leonhardi Euleri Opera omnia. Series quarta A: Commercium epistolicum Volumen secundum: Commercium cum Johanne (I) Bernoulli et Nicolao (I) Bernoulli. Birkhäuser, Basel
Ferraro, G. 2008. The rise and development of the theory of series up to the early 1820s. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag.
Fuss, P.-H., editor. 1843. Correspondance mathématique et physique de quelques célèbres géomètres du XVIII ème siècle. Tomes I, II. L’Académie Impériale des Sciences de Saint-Pétersbourg, St.-Pétersbourg. The Sources of Science, No. 35, Johnson Reprint Corporation, New York and London, 1968.
Gautschi W. (2008) On Euler’s attempt to compute logarithms by interpolation: A commentary to his letter of February 16, 1734 to Daniel Bernoulli. Journal of Computational and Applied Mathematics 219(2): 408–415
Gillispie, C.C. editors. 1970–1980. Dictionary of Scientific Biography. New York: Charles Scribner’s Sons.
Hardy, G.H. 1949. Divergent series. Oxford University Press.
Hardy, G.H., and Ramanujan, S. 1918. Asymptotic formulae in combinatory analysis. Proceedings of the London Mathematical Society, Series 2 17: 75–115. Table IV calculated by Percy A. MacMahon.
Hardy, G.H., and Wright, E.M. 1980. An Introduction to the theory of numbers. Oxford University Press, fifth edition.
Hofmann J.E. (1959) Um Eulers erste Reihenstudien. In: Schröder K. (eds) Sammelband der zu Ehren des 250 Geburtstages Leonhard Eulers der Deutschen Akademie der Wissenschaften zu Berlin vorgelegten Abhandlungen. Akademie-Verlag, Berlin, pp 139–208
Iwaniec, H. 1997. Topics in classical automorphic forms, vol. 17 of Graduate Studies in Mathematics. American Mathematical Society.
Juškevič, A.P., Smirnov, V.I., Habicht, W. (eds) (1975) Leonhardi Euleri Opera omnia. Series quarta A, Birkhäuser
Juškevič, A.P., and Taton, R., editors. 1980. Leonhardi Euleri Opera omnia. Series quarta A: Commercium epistolicum. Volumen quintum: Correspondance de Leonhard Euler avec A. C. Clairaut, J. d’Alembert et J. L. Lagrange. Basel: Birkhäuser.
Juškevič, A.P., and Winter, E. (eds.). 1961. Die Berliner und die Petersburger Akademie der Wissenschaften im Briefwechsel Leonhard Eulers. Teil 2: Der Briefwechsel L. Eulers mit Nartov, Razumovskij, Schumacher, Teplov und der Petersburger Akademie 1730–1763. Quellen und Studien zur Geschichte Osteuropas. Berlin: Akademie-Verlag.
Juškevič, A.P., and Winter, E. (eds.) 1965. Leonhard Euler und Christian Goldbach. Briefwechsel 1729–1764. Berlin: Akademie-Verlag. Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin. Jahrgang 1965 Nr. 1.
Juškevič, A.P., and Winter, E., editors. 1976. Die Berliner und die Petersburger Akademie der Wissenschaften im Briefwechsel Leonhard Eulers. Teil 3: Wissenschaftliche und wissenschaftsorganisatorische Korrespondenzen 1726–1774. Quellen und Studien zur Geschichte Osteuropas. Berlin: Akademie-Verlag.
Kempner A.J. (1923) The development of “partitio numerorum,” with particular reference to the work of Messrs. Hardy, Littlewood and Ramanujan. American Mathematical Monthly 30(7): 354–369
Kiselev A.A., Matvievskaja G.P. (1965) Neopublikovannye zapisi Eilera po partitio numerorum. Istoriko-Matematicheskie Issledovaniya 16: 145–180
Knobloch E. (1974) The mathematical studies of G. W. Leibniz on combinatorics. Historia Mathematica 1(4): 409–430
Knobloch, W., editor. 1984. Leonhard Eulers Wirken an der Berliner Akademie der Wissenschaften 1741–1766. Berlin: Akademie-Verlag. Studien zur Geschichte der Akademie der Wissenschaften der DDR. Band 11.
Koelink E., Van Assche W. (2009) Leonhard Euler and a q-analogue of the logarithm. Proceedings of the American Mathematical Society 137(5): 1663–1676
Legendre, A.-M. 1830. Théorie des nombres. Tome II. Firmin-Didot, Paris, third edition.
Markushevich, A.I. 1967. Theory of functions of a complex variable, vol. III. Prentice-Hall. Translated from the Russian. Revised English edition translated and edited by Richard A. Silverman.
Mikhajlov G.K. (1959) Notizen über die unveröffentlichten Manuskripte von Leonhard Euler. In: Schröder K. (eds) Sammelband der zu Ehren des 250 Geburtstages Leonhard Eulers der Deutschen Akademie der Wissenschaften zu Berlin vorgelegten Abhandlungen. Akademie-Verlag, Berlin, pp 256–280
Nevskaja, N.I. (eds) (2000) Letopis’ Rossiiskoi Akademii Nauk. Nauka, St. Petersburg
Patterson S.J. (2007) Gauss sums. In: Goldstein C., Schappacher N., Schwermer J. (eds) The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae. Springer-Verlag, Berlin, pp 505–528
Pólya, G. 1954. Mathematics and plausible reasoning. Induction and analogy in mathematics, vol. I. Princeton University Press.
Rademacher H. (1940) Fourier expansions of modular forms and problems of partition. Bulletin of the American Mathematical Society 46: 59–73
Rademacher, H. (1969). Comments on Euler’s “De mirabilibus proprietatibus numerorum pentagonalium”. In Number theory and analysis (papers in Honor of Edmund Landau), pages 257–268. New York: Plenum.
Rademacher, H. 1973. Topics in analytic number theory. Springer-Verlag, New York. Edited by Grosswald, E., Lehner, J., and Newman, M. Die Grundlehren der mathematischen Wissenschaften, Band 169.
Rudio, F., editor. 1915. Leonhardi Euleri Opera omnia, series I, volumen II: Leonhardi Euleri Commentationes arithmeticae, volumen primum. B. G. Teubner, Leipzig and Berlin.
Sandifer, C.E. 2007a. Some facets of Euler’s work on series. In Bradley, R.E., and Sandifer, C.E., editors. Leonhard Euler: Life, Work and Legacy, pages 279–302. Elsevier. Studies in the History and Philosophy of Mathematics, vol. 5.
Sandifer, C.E. 2007b. The early mathematics of Leonhard Euler. The MAA Tercentenary Euler Celebration, vol. 1. Spectrum Series. The Mathematical Association of America.
Scharlau, W. 1983. Eulers Beiträge zur partitio numerorum und zur Theorie der erzeugenden Funktionen. In: Burckhardt, J.J., Fellmann, E.A., and Habicht, W., editors. Leonhard Euler 1707–1783. Beiträge zu Leben und Werk. Gedenkband des Kantons Basel-Stadt, pages 135–150. Basel: Birkhäuser.
Scriba C.J. (1984) Eulers zahlentheoretische Studien im Lichte seines wissenschaftlichen Briefwechsels. In: Knobloch E., Louhivaara I.S., Winkler J. (eds) Zum Werk Leonhard Eulers. Vorträge des Euler-Kolloquiums im Mai 1983 in Berlin. Basel, Birkhäuser, pp 67–94
Segal, S.L. 2008. Nine introductions in complex analysis, vol. 208 of North-Holland Mathematics Studies. Elsevier, revised edition.
Smirnov, V.I. (eds) (1963) Pis’ma k uchenym. Nauka, Moscow-Leningrad
Stäckel, P., and Ahrens, W. 1908. Der Briefwechsel zwischen C. G. J. Jacobi und P. H. von Fuss über die Herausgabe der Werke Leonhard Eulers. B. G. Teubner, Leipzig.
Struik, D.J., editor. 1969. A source book in mathematics, 1200-1800. Source Books in the History of the Sciences. Harvard University Press.
Tannery, P., Henry, C. (eds) (1891) Oeuvres de Fermat. Tome I. Gauthier-Villars et fils, Paris
Titchmarsh, E.C. 1939. The theory of functions. Oxford University Press, second edition.
van der Waerden, B.L., editor. 1975. Die Werke von Jakob Bernoulli. Band 3. Wahrscheinlichkeitsrechnung. Birkhäuser, Basel. Herausgegeben von der Naturforschenden Gesellschaft in Basel.
Weil, A. 1976. Elliptic functions according to Eisenstein and Kronecker, vol. 88 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag.
Weil, A. 1984. Number theory: an approach through history from Hammurapi to Legendre. Birkhäuser.
Winter, E. (eds) (1957) Die Registres der Berliner Akademie der Wissenschaften 1746–1766. Dokumente für das Wirken Leonhard Eulers in Berlin. Akademie-Verlag, Berlin
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Niccolò Guicciardini.
Part of this work was done at the School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada.
Rights and permissions
About this article
Cite this article
Bell, J. A summary of Euler’s work on the pentagonal number theorem. Arch. Hist. Exact Sci. 64, 301–373 (2010). https://doi.org/10.1007/s00407-010-0057-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00407-010-0057-y