Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas

Tatiana Lavrinenko

Revue d'histoire des mathématiques (2002)

  • Volume: 8, Issue: 1, page 67-112
  • ISSN: 1262-022X

Abstract

top
This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.

How to cite

top

Lavrinenko, Tatiana. "Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas." Revue d'histoire des mathématiques 8.1 (2002): 67-112. <http://eudml.org/doc/252068>.

@article{Lavrinenko2002,
abstract = {This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.},
author = {Lavrinenko, Tatiana},
journal = {Revue d'histoire des mathématiques},
keywords = {diophantine equations; algebraic geometry; elliptic curve; rational point; Lucas; Sylvester; story; Diophantine equations; Story},
language = {eng},
number = {1},
pages = {67-112},
publisher = {Société mathématique de France},
title = {Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas},
url = {http://eudml.org/doc/252068},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Lavrinenko, Tatiana
TI - Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas
JO - Revue d'histoire des mathématiques
PY - 2002
PB - Société mathématique de France
VL - 8
IS - 1
SP - 67
EP - 112
AB - This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.
LA - eng
KW - diophantine equations; algebraic geometry; elliptic curve; rational point; Lucas; Sylvester; story; Diophantine equations; Story
UR - http://eudml.org/doc/252068
ER -

References

top
  1. [1] BASHMAKOVA ( Isabella G.) [1968] Diophante et Fermat, Revue d’histoire des sciences et de leurs applications, 19 (1968), pp. 283–306. 
  2. [2] BASHMAKOVA ( Isabella G.) [1981] Arithmetic of Algebraic Curves from Diophantus to Poincaré, Historia mathematica, 8 (1981), pp. 393–416. Zbl0471.01003MR635360
  3. [3] CAUCHY ( Augustin-Louis) [Œuvres] Œuvres complètes, 27 vols. (two series), Paris: Gauthier-Villars, 1882–1974. Zbl1204.00001MR472451
  4. [4] CAUCHY ( Augustin-Louis) [1826] Exercices de mathématiques, Paris, 1826; Œuvres (II) 6, pp. 286–315. 
  5. [5] CLEBSCH ( Rudolf Friedrich Alfred) [1864] Über einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung, Journal für die reine und angewandte Mathematik, 63 (1864), pp.94–121 . 
  6. [6] COOKE ( Roger) & RICKEY ( V.Frederick) [1989] W.E. Story of Hopkins and Clark, in Duren (Peter) et al., eds., A Century of Mathematics in America – Part III, Providence: American Mathematical Society, 1989, pp. 29–76. MR1025341
  7. [7] DAHAN-DALMÉDICO ( Amy) & PEIFFER ( Jeanne) [1982] Une histoire des mathématiques. Routes et dédales, Paris: Études vivantes, 1982 ; 2e éd., Paris : Éditions du Seuil, 1986, 314p. 
  8. [8] DAHAN-DALMÉDICO ( Amy) & PEIFFER ( Jeanne) [1986] Routes et dédales (Russian translation), Moscow : MIR, 1986. MR890326
  9. [9] DESBOVES ( Alain) [1879] Mémoire sur la résolution en nombres entiers de l’équation a X m + b Y m = c Z n , Nouvelles annales de mathématiques, 2e série, 18 (1879), pp.265–279, 398–410, 433–444, 481–499. JFM11.0138.04
  10. [10] DÉCAILLOT ( Anne-Marie) [1998] L’arithméticien Édouard Lucas (1842–1891): théorie et instrumentation, Revue d’histoire des mathématiques, 4 (1998), pp.191–236. Zbl0938.01035MR1716017
  11. [11] DICKSON ( Leonard Eugene) [1920] History of the Theory of Numbers, vol.2: Diophantine Analysis, Washington : Carnegie Institute of Washington, 1920 (reprint. New York : Chelsea, 1952). Zbl1214.11002
  12. [12] DIEUDONNÉ ( Jean), éd. [1978] Abrégé d’histoire des mathématiques, 1700–1900, t.I–II, Paris: Hermann, 1978. Zbl0656.01001
  13. [13] ELLISON ( W. & F.) [1978] Théorie des nombres, [Abrégé], I, chap.V, pp.165–334. 
  14. [14] EULER ( Leonhard) [Opera] Leonhardi Euleri Opera omnia, ser.1, 29vol., Leipzig-Berlin: Teubner, 1911–1956. 
  15. [15] EULER ( Leonhard) [1770] Vollständige Anleitung zur Algebra, Zweyter Teil, Petersbourg, 1770; Opera (I), 1, pp.209–498. Zbl0083.24406
  16. [16] FERMAT ( Pierre) [1670] Doctrinae analyticae inventum novum (compiled by J.de Billy from Fermat’s letters); Œuvres de Pierre Fermat, t.1, Paris: Blanchard, 1999, pp.157–232. 
  17. [17] GOLDSTEIN ( Catherine) [1995] Un théorème de Fermat et ses lecteurs, Saint-Denis: PUV, 1995. Zbl0879.01013MR1351497
  18. [18] HARKIN ( Duncan) [1957] On the Mathematical Work of François-Édouard-Anatole Lucas, L’Enseignement mathématique, 2e série, 3 (1957), pp.276–288. Zbl0078.00403MR97292
  19. [19] HOFMANN ( Joseph Ehrenfried) [1961] Über zahlentheoretische Methoden Fermats und Eulers, ihre Zusammenhänge und ihre Bedeutung, Archive for History of Exact Sciences, 1 (1961), pp.122–159. Zbl0094.00302MR124987
  20. [20] HURWITZ ( Adolf) [1917] Über ternäre diophantische Gleichungen dritten Grades, Vierteljahresschrift der Naturforschenden Gesellschaft in Zürich, 62 (1917), pp.207–229. Zbl46.0205.05JFM46.0205.05
  21. [21] JACOBI ( Carl Gustav Jacob) [1835] De usu theoriae integralium ellipticorum et integralium abelianorum in analysi diophantea, J. reine angew. Math., 13 (1835), pp. 353–355. 
  22. [22] KAUCHIKAS ( Algerdas) [1979] Diophantus and Indeterminate Analysis in the Works of European Mathematicians of the 13th–16th Centuries, Dissertation, Moscow, 1979 (in Russian). 
  23. [23] LAGRANGE ( Joseph Louis) [Œuvres] Œuvres de Lagrange, J.-A.Serret and G.Darboux, eds., 14vol., Paris: Gauthiers-Villars, 1867–1892. 
  24. [24] LAGRANGE ( Joseph Louis) [1777] Sur quelques problèmes de l’analyse de Diophante, Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, 1777; Œuvres 4, pp.377–398. 
  25. [25] LAVRINENKO ( Tatiana A.) [1982] Solving of Indeterminate Equations of Third and Forth Degrees in Rational Numbers in the 19th Century, Moscow: deponir. VINITI AN SSSR, No. 3669–83, 1982 (in Russian). 
  26. [26] LAVRINENKO ( Tatiana A.) [1983] Solving of Indeterminate Equations of Third and Fourth Degrees in the Late Euler’s Works, Istoriko-Matematicheskie Issledovaniya, 27 (1983), pp.67–79 (in Russian). Zbl0523.01006
  27. [27] LAVRINENKO ( Tatiana A.) [1985] On Methods of Solving Indeterminate Equations in Rational Numbers in the 18th–19th centuries, Istor.-Mat. Issledov., 28 (1985), pp.202–223 (in Russian). Zbl0583.01006MR823885
  28. [28] LAVRINENKO ( Tatiana A.) [1988] Diophantine Equations in L.Euler’s Works, in The Development of Leonard Euler’s Ideas and Modern Science, Moscow: Nauka, 1988, pp.153–165 (in Russian). MR952163
  29. [29] LUCAS ( Édouard) [1873] Recherches sur l’analyse indéterminée et l’arithmétique de Diophante, Moulins: Desrosiers, 1873; reprint Paris: Blanchard, 1961. Zbl0094.02602
  30. [30] LUCAS ( Édouard) [1875] Questions d’analyse indéterminée, Nouv. ann. math., 2e série, 14 (1875), p. 526. JFM07.0099.03
  31. [31] LUCAS ( Édouard) [1877] Recherches sur plusieurs ouvrages de Léonard de Pise et sur diverses questions d’arithmétique supérieure, Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, 10 (1877), pp.129–193, 239–293. JFM09.0111.02
  32. [32] LUCAS ( Édouard) [1878] Sur l’analyse indéterminée du troisième degré et sur la question 802 (Sylvester), Nouv. ann. math., 2e série, 17 (1878), pp.507–514. JFM10.0147.01
  33. [33] LUCAS ( Édouard) [1879] Sur l’analyse indéterminée du troisième degré. Démonstration de plusieurs théorèmes de M.Sylvester, American Journal of Mathematics, 2 (1879), pp.178–185. JFM11.0135.01
  34. [34] NEWTON ( Isaac) [1971] The Mathematical Papers of Isaac Newton, vol.4, ed. D.T.Whiteside, Cambridge, 1971. Zbl0215.04201
  35. [35] PARSHALL ( Karen Hunger) & ROWE ( David E.) [1994] The Emergence of the American Mathematical Research Community 1876-1900: J.J.Sylvester, Felix Klein, and E.H.Moore, Providence: American Mathematical Society, 1994. Zbl0802.01005MR1290994
  36. [36] PARSHALL ( Karen Hunger) [1998] James Joseph Sylvester: Life and Work in Letters, Oxford : Oxford University Press, 1998. Zbl0929.01018MR1674190
  37. [37] POINCARÉ ( Jules Henri) [1901] Sur les propriétés arithmétiques des courbes algébriques, Journal de mathématiques pures et appliquées, 5e série, 7 (1901), pp.161–233. JFM32.0564.06
  38. [38] RASHED ( Roshdi), éd. [1984] Diophante, Les Arithmétiques, Paris : Les Belles Lettres, 1984 (introduction, notamment). 
  39. [39] SALMON ( George) [1879] A Treatise on the Higher Plane Curves:Intended as a Sequel to a Treatise on Conic Sections , 3d ed., Dublin, 1879. Zbl0056.14202JFM05.0340.03
  40. [40] SCHAPPACHER ( Norbert) [1991] Développement de la loi de groupe sur une cubique, dans Séminaire de théorie des nombres, Paris 1988/1989 (Progress in Mathematics 91), Boston: Birkhäuser, 1991, pp.159–184. Zbl0742.14019MR1104705
  41. [41] SCHLESINGER ( Ludwig) [1909] Über ein Problem der Diophantischen Analysis bei Fermat, Euler, Jacobi und Poincaré, Jahresbericht der Deutschen Mathematiker-Vereinigung, 17 (1909), pp.57–67. JFM39.0258.01
  42. [42] STORY ( William E.) [1880] On the Theory of Rational Derivation on a Cubic Curve, Amer. J. Math., 3 (1880), pp.356–387. MR1505277JFM13.0084.01
  43. [43] SYLVESTER ( James Joseph) [Mathematical Papers] Collected Mathematical Papers, 4 vols., Cambridge : Cambridge University Press, 1904–1911. JFM43.0026.01
  44. [44] SYLVESTER ( James Joseph) [1847a] An Account of a Discovery in the Theory of Numbers Relative to the Equation A x 3 + B y 3 + C z 3 = D x y z , Philosophical Magazine, 31 (1847), pp. 189–191; Mathematical Papers, 1, pp.107–109. 
  45. [45] SYLVESTER ( James Joseph) [1847b] On the Equation in Numbers A x 3 + B y 3 + C z 3 = D x y z and Its Associate System of Equations, Phil. Mag., 31 (1847), pp.293–296; Mathematical Papers, 1, pp.110–113. 
  46. [46] SYLVESTER ( James Joseph) [1847c] On the General Solution (in Certain Cases) of the Equation x 3 + y 3 + A z 3 = M x y z , etc., Phil. Mag., 31 (1847), pp.467–471; Mathematical Papers, 1, pp.114–118. 
  47. [47] SYLVESTER ( James Joseph) [1856] Recherches sur les solutions en nombres entiers positifs ou négatifs de l’équation cubique homogène à trois variables, Annali di scienze matematiche e fisiche, 7 (1856), pp.398–400; Mathematical Papers, 2, pp.63–64. 
  48. [48] SYLVESTER ( James Joseph) [1858] Note on the Algebraical Theory of Derivative Points of Curves of the Third Degree, Phil. Mag., 16 (1858), pp.116–119; Mathematical Papers, 2, pp.107–109. 
  49. [49] SYLVESTER ( James Joseph) [1867] Question no 802, Nouv. ann. math., 2e série, 6 (1867), p.96. 
  50. [50] SYLVESTER ( James Joseph) [1879/80] On Certain Ternary Cubic-Form Equations, Amer. J. Math., 2 (1879), pp.280–285 and 357–393; 3 (1880), pp.58–88 and 179–189; Mathematical Papers, 3, pp.312–391. Zbl11.0141.01MR1505225
  51. [51] WEIL ( André) [1983] Number Theory. An Approach through History: from Hammurapi to Legendre, Boston, etc.: Birkhäuser, 1983. Zbl0531.10001MR734177

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.