Fagnano's method for solving algebraic equations: Its historical overview and development

A.N. Abyzov

Kazan Federal University, Kazan, 420008 Russia

E-mail: aabyzov@kpfu.ru

Received September 16, 2021

 

ORIGINAL ARTICLE

Full text PDF

DOI: 10.26907/2541-7746.2021.3-4.304-348

For citation: Abyzov A.N. Fagnano's method for solving algebraic equations: Its historical overview and development. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, vol. 163, no. 3–4, pp. 304–348. doi: 10.26907/2541-7746.2021.3-4.304-348. (In Russian)

Abstract

In 1750, Giulio Carlo Fagnano dei Toschi's treatise “Produzioni matematiche” was published in two volumes. The second volume of the treatise contains a work in which Fagnano proposed a uniform method for solving algebraic equations up to the fourth degree. Fagnano's method, as we call it in this article, is based on a comparison of algebraic equations with identities arising in the representation of an expression of the form (a1 + ... + am)n in terms of lesser powers of a1 + ... + am. Here we explore the results related to the use of certain types of algebraic identities in solving a number of families of algebraic equations that are solvable by radicals. Various connections of these identities with linear recurrent sequences and trigonometric identities are considered. A historical survey devoted to the use of Fagnano's method for solving algebraic equations is presented in section 1. Section 2 is devoted to algebraic equations that are solvable by radicals closely related to Chebyshev polynomials and whose solution by radicals is based on the use of the Kummer identity. In section 3, Fagnano's method is used to study some families of algebraic equations that are solvable by radicals. In sections 4 and 5, the connections of the identities considered in the previous sections with the well-known linear recurrent sequences are analyzed. In section 6, a connection is established between the identities covered in this article and the group determinants.

Keywords: algebraic equations, Lucas sequences, group determinants

Acknowledgments. D.T. Tapkin's continuous support during the study and helpful advice are gratefully acknowledged.

References

  1. Fagnano G.C. Soluzione di quattro problemi analitici. In: Fagnano G.C. Opere Matematiche. T. 2. Milano, Roma, Napoli, Albrighi Segati, 1911. 516 p. (In Italian)
  2. Wieleitner H. Istoriya matematiki ot Dekarta do serediny XIX stoletiya [History of Mathematics from Descartes to the Mid-19th Century]. Moscow, GIFML, 1960. 468 p. (In  Russian)
  3. Dulaurens Fr. Specimina Mathematica. Paris, 1667.
  4. Schneider I. Der Mathematiker Abraham de Moivre (1667–1754). Arch. Hist. Exact Sci., 1968, vol. 5, no. 3/4, pp. 177–317. (In German)
  5. Haukkanen P., Merikoski J., Mustonen S. Some polynomials associated with regular poly- gons. Acta Univ. Sapientiae. Math., 2014, vol. 6, no. 2, pp. 178–193. doi: 10.1515/ausm- 2015-0005.
  6. Savio D.Y., Suryanarayan E.R. Chebychev polynomials and regular polygons. Am. Math. Mon., 1993, vol. 100, no. 7, pp. 657–661. doi: 10.1080/00029890.1993.11990466.
  7. Laughlin J.Mc. Combinatorial identities deriving from the n-th power of a 2 × 2 matrix. Integers, 2004, vol. 4, pp. 1–15.
  8. Eulero L. De resolutione aequationum cuiusvis gradus. Novi Comment. Acad. Sci. Imp. Petropolitanae. Petropolis, 1764, vol. 9 (1762–1763), pp. 70–98. (In Latin)
  9. Kaddoura I., Mourad B. On a class of matrices generated by certain generalized  permutation matrices and  applications. Linear Multilinear Algebra, 2018, vol. 67,  no.  10, pp. 2117–2134. doi: 10.1080/03081087.2018.1484420.
  10. Grunert J.A. Die allgemeine Cardanische Formel. Arch. Math. Phys., 1863, Bd. 40, S. 246–249. (In German)
  11. Blum-Smith B., Wood J. Chords of an ellipse, Lucas polynomials, and cubic equations. Am. Math. Mon., 2020, vol. 127, no. 8, pp. 688–705. doi: 10.1080/00029890.2020.1785253.
  12. Solomon R. Abstract Algebra. Belmont, Thompson Brooks/Cole, 2003. 227 p.
  13. Spearman B.K., Williams K.S. DeMoivre's quintic and a theorem of Galois. Far East J. Math. Sci., 1999, vol. l, pp. 137–143.
  14. Borger R.L. On De Moivre's quantic. Am. Math. Mon., 1908, vol. 15, no. 10, pp. 171–174. doi: 10.1080/00029890.1908.11997448.
  15. Maistrova A.L. Solving algebraic equations in L. Euler's works. In: Istoriko-matematicheskie issledovaniya [Historical and Mathematical Studies]. Yushkevich A.P. (Ed.). Moscow, Nauka, 1985, no. 29, pp. 189–199. (In Russian)
  16. Valles F. Des formes imaginaires en algebra. Paris,  Gauthier-Villars,  1873.  539  p. (In  French)
  17. Stedall J.A. From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra. Zürich, Eur. Math. Soc., 2011. 224 p.
  18. Dickson L.E. History of the Theory of Numbers. Vol. I: Divisibility and primality. Washington, Carnegie Inst., 1919. XII, 486 p.
  19. Girard A. Invention nouvelle en l'algebre. Amsterdam, 1629. 72 p.
  20. Waring E. Miscellanea analytica de aequationibus algebraicis, et curvarum proprietatibus. Cambridge, Thurlbourn Woodyer, 1762. 206 p. (In Latin)
  21. Sushkevich A.K. Osnovy vysshei algebry [Fundamentals of Higher Algebra]. Moscow, Leningrad, Gostekhizdat, 1941. 460 p. (In Russian)
  22. Lobachevsky N.I. Essays on algebra. Algebra, or the calculus of finite quantities. Lowering the degree in a binominal equation when the exponent minus a unit is a multiple of 8. In: Lobachevsky N.I. Polnoe sobranie sochinenii [Complete Collection of Works]. Vol. 4. Kagan V.F. (Ed.). Moscow, Leningrad, Gostekhizdat, 1948. 472 p. (In Russian)
  23. De Moivre A. Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, et superiorum, ad infinitum usque pergendo, in terminis finitis, ad instar regularum pro cubicis quae vocantur Cardani resolutio analytica. Philos. Trans., 1707, vol. 25, no. 309, pp. 2368–2371. (In Latin)
  24. Wituła R., Słota D. Cardano's formula, square roots, Chebyshev polynomials and radicals. J. Math. Anal. Appl., 2010, vol. 363, no. 2, pp. 639–647. doi: 10.1016/j.jmaa.2009.09.056.
  25. Dickson L.E. Elementary Theory of Equations. New York, Wiley, 1914. 196 p.
  26. Dickson L.E. Linear Groups: With an Exposition of the Galois Field Theory. Leipzig, Teubner,  1901.  X,  312  p.
  27. Williams K.S. A generalization of Cardan's solution of the cubic. Math. Gaz., 1962, vol. 46, no. 357, pp. 221–223.
  28. Lehmer D.H. On the multiple solutions of the Pell equation. Ann. Math., 1928, vol. 30, no. 1/4, pp. 66–72.
  29. Euler L. Vvedenie v analiz beskonechnykh [Introduction to Analysis of the Infinite]. Vol. 1. Moscow, Fizmatgiz, 1961. 315 p. (In Russian)
  30. Wituła R., Słota  D. Cauchy, Ferrers–Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences. Cent. Eur. J. Math., 2006, vol. 4, no. 3, pp. 531–546. doi: 10.2478/s11533-006-0022-9.
  31. Kepler J. Harmonice Mundi. In: Kepler J. Opera Omnia. Bd. 5. Frankfurt, Heyder Zim- mer, 1864, S. 75–327.
  32. Kappraff J. Beyond Measure: A Guided Tour through Nature, Myth, and Number. River Edge, N. J., World Sci. Publ., 2002. XXX, 582 p.
  33. Gauss C.F. Trudy po teorii chisel [Works on the Number Theory]. Moscow, Izd. Akad. Nauk SSSR, 1959. 980 p. (In Russian)
  34. Eisenstein G. Application de l'algebre a l'arithmetique transcendante. J. Reine Angew. Math., 1845, T. 29, pp. 177–184. doi: 10.1515crll. (In French)
  35. Liouville J. Sur la loi de reciprocite dans la theorie des residus quadratiques. J. Math. Pures Appl., 1847, T. 12, pp. 95–96. (In French)
  36. Baumgart O., Lemmermeyer F. The Quadratic Reciprocity Law: A Collection of Classical Proofs. Cham, Heidelberg, New York, Dordrecht, London,  Birkhauser  Springer,  2015. XIV, 172 p.
  37. Perrin R. Item 1484. L'Intermed Math., 1899, T. 6, pp. 76–77. (In French)
  38. Minton G.T. Three approaches to a sequence problem. Math. Mag., 2011, vol. 84, no. 1, pp. 33–37. doi: 10.4169/math.mag.84.1.033.
  39. Schönemann T. Theorie der symmetrischen Functionen der Wurzeln einer Gleichung. Allgemeine Sätze über Congruenzen nebst einigen Anwendungen derselben. J. Reine Angew. Math., 1839, Bd. 19, S. 289–308. (In German)
  40. Shatunovskii S.O. Ob usloviyakh sushchestvovaniya n neravnykh kornei sravneniya n-i stepeni po prostomu modulyu [On the Conditions for the Existence of n Unequal Roots of Congruence of n-Degree Using a Simple Module]. Kazan, Tip.-Lit. Imp. Univ., 1903. 19 p. (In Russian)
  41. Arnold V.I. On the matricial version of Fermat–Euler congruences. Jpn. J. Math., 2006, vol. 1, art. 1, pp. 1–24. doi: 10.1007/s11537-006-0501-6.
  42. Browder F.E. The Lefschetz fixed point theorem and asymptotic fixed point theorems in partial differential equations and related topics. In: Lecture Notes in Mathematics. Vol. 446. New York, Springer-Verlag, 1975, pp. 96–122.
  43. Peitgen H.O. On the Lefschetz number for iterates of continuous mappings. Proc. Am. Math. Soc., 1976, vol. 54, no. 1, pp. 441–444. doi: 0.1090/S0002-9939-1976-0391074-1.
  44. Janichen W. Uber die Verallgemeinerung einer Gaussschen Formel aus der Theorie der hoheren Kongruenzen. Sitzungsber. Berlin. Math. Ges., 1921, Bd. 20, S. 23–29. (In German)
  45. Adams W.W., Shanks D. Strong primality tests that are not sufficient. Math. Comput., 1982, vol. 39, no. 159, pp. 255–300.
  46. Vinberg E.B. Fermat's little theorem and generalizations. Mat. Prosveshchenie. Ser. 3, 2008, no. 12, pp. 43–53. (In Russian)
  47. Zarelua A.V. On matrix analogs of Fermat's little theorem. Math. Notes, 2006, vol. 79, no. 6, pp. 783–796. doi: 10.1007/s11006-006-0090-y.
  48. Jezierski J., Marzantowicz W. Homotopy methods in topological fixed and periodic points theory. In: Gorniewicz L. (Ed.) Topological Fixed Point Theory and Its Applications. Vol. 3. Dordrecht, Springer, 2006. XII, 319 p.
  49. Schur I. Arithmetische Eigenschaften der Potenzsummen einer algebraischen Gleichung. Compos. Math., 1937, Bd. 4, S. 432–444. (In German)
  50. Smyth C.J. A coloring proof of a generalisation of Fermat's little theorem. Am. Math. Mon., 1986, vol. 93, no. 6, pp. 469–471.
  51. Steinlein H. Fermat's little theorem and Gauss congruence: Matrix versions and cyclic permutations. Am. Math. Mon., 2017, vol. 124, no. 6, pp. 548–553.
  52. Zarelua A.V. On congruences for the traces of powers of some matrices. Proc. Steklov Inst. Math., 2008, vol. 263, pp. 78–98. doi: 10.1134/S008154380804007X.
  53. Beukers F., Houben M.R., Straub A. Gauss congruences for rational functions in several variables. Acta Arithmetica, 2018, vol. 184, no. 4, pp. 341–362. doi: 10.4064/aa170614-13-7.
  54. Minton G.T. Linear recurrence sequences satisfying congruence conditions. Proc.  Am. Math. Soc., 2014, vol. 142, no. 7, pp. 2337–2352.
  55. Chebotarev N.G. Samuil Osipovich Shatunovskii (on the occasion of the 10th anniversary of his death). Usp. Mat. Nauk, 1940, no. 7, pp. 316–321. (In Russian)
  56. Lambek J. Kol'tsa i moduli [Lectures on Rings and Modules]. Moscow, Mir, 1971. 280 p. (In Russian)
  57. Sun Z.-H. Cubic and quartic  congruences  modulo  a  prime.  J. Number Theory,  2003, vol. 102, no. 1, pp. 41–89. doi: 10.1016/S0022-314X(03)00067-2.
  58. Frobenius G. Teoriya kharakterov i predstavlenii [Theory of Group Characters and Representations]. Kharkiv, ONTI, 1937. 214 p. (In Russian)
  59. Johnson K.W. Group Matrices, Group Determinants  and  Representation  Theory: The Mathematical Legacy of Frobenius. New York, Springer Int. Publ., 2019. 409 p. (Lecture Notes in Mathematics. Vol. 2233)
  60. Pen-Tung Sah A. A uniform method of solving cubics and quartics. Am. Math. Mon., 1945, vol. 52, no. 4, pp. 202–206.

 

The content is available under the license Creative Commons Attribution 4.0 License.