Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Integer points on $ y\sp{2}=x\sp{3}-7x+10$


Authors: Andrew Bremner and Nicholas Tzanakis
Journal: Math. Comp. 41 (1983), 731-741
MSC: Primary 11D25; Secondary 14K07
DOI: https://doi.org/10.1090/S0025-5718-1983-0717717-X
MathSciNet review: 717717
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The 26 integer solutions of $ {y^2} = {x^3} - 7x + 10$ are found and an error in a published table of fundamental units is corrected.


References [Enhancements On Off] (What's this?)

  • [1] W. E. H. Berwick, "Algebraic number fields with two independent units," Proc. London Math. Soc. v. 34, 1932, pp. 360-378.
  • [2] G. Billing, "Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlecht eins," Nova Acta Reg. Soc. Scient. Upsaliensis, Ser. IV, v. 11, 1938.
  • [3] B. J. Birch & W. Kuyk (eds.), Modular Functions of One Variable IV, Proc. Internat. Summer School, Antwerp 1972, Springer, 1975. MR 0376533 (51:12708)
  • [4] B. J. Birch & H. P. F. Swinnerton-Dyer, "Notes on elliptic curves I," J Reine Angew. Math., v. 212, 1963, pp. 7-25; "Notes on elliptic curves II," ibid., v. 218, 1965, pp. 79-108. MR 0146143 (26:3669)
  • [5] A. J. Brentjes, "A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields", J. Reine Angew. Math., v. 326, 1981, pp. 18-44. MR 622343 (83b:10037)
  • [6] J. W. S. Cassels, "Diophantine equations with special reference to elliptic curves," J. London Math. Soc., v. 41, 1966, pp. 193-291. MR 0199150 (33:7299)
  • [7] F. B. Coghlan & N. M. Stephens, "The Diophantine equation $ {x^3} - {y^2} = k$," Computers in Number Theory (Atkin and Birch, eds.), Proc. Atlas Symp. No. 2, 1969, Academic Press, 1971. MR 0314733 (47:3285)
  • [8] B. N. Delone & D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Transl. Math. Monographs, Vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR 0160744 (28:3955)
  • [9] R. Finkelstein & H. London, "On Mordell's equation $ {y^2} - k = {x^3}$: An interesting case of Sierpinski," J. Number Theory, v. 2, 1970, pp. 310-321. MR 0268120 (42:3019)
  • [10] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • [11] S. Lang, Elliptic Curves, Diophantine Analysis, Springer, 1978. MR 518817 (81b:10009)
  • [12] S. Lang, Conjectured Diophantine Estimates on Elliptic Curves, Shafarevich's 60th Birthday Volume, Birkhauser, 1983. MR 717593 (85d:11024)
  • [13] W. Ljunggren, "On the Diophantine equation $ {y^2} - k = {x^3}$, Acta Arith., v. 8, 1963, pp. 451-463. MR 0158859 (28:2082)
  • [14] J.-F. Mestre, "Construction d'une courbe elliptique de rang $ \geqslant 12$," Comptes Rendus, v. 295, 1982, pp. 643-644. MR 688896 (84b:14019)
  • [15] L. J. Mordell, Diophantine Equations, Academic Press, 1969. MR 0249355 (40:2600)
  • [16] G. Sansone, "I punti di coordinate rationali e, in particolare, di coordinate intere della cubica ellittica $ {y^2} = {x^3} - x + 1$," Ann. Mat. Pura Appl. (4), v. 125, 1980, pp. 1-11.
  • [17] Th. Skolem, "The use of a p-adic method in the theory of diophantine equations," Bull. Soc. Math. Belg., v. 7, 1955, pp. 83-95. MR 0072155 (17:237h)
  • [18] Th. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen, 8de Skand. mat. Kongr., Stockholm, 1934.
  • [19] J. Tate, "Algorithm for determining the type of a singular fiber in an elliptic pencil," Modular Functions of One Variable IV (Birch and Kuyk, eds.). Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33-52. MR 0393039 (52:13850)
  • [20] N. Tzanakis, "The Diophantine equation $ {x^3} - 3x{y^2} - {y^3} = 1$ and related equations," J. Number Theory. (To appear.)
  • [21] A. Wiman, "Über die Punkte mit ganzzahligen Koordinaten auf gewissen Kurven dritter Ordnung, 12te Skand. Matematikerkongressen, Lund, 1953, pp. 317-323 (1954). MR 0065585 (16:450a)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11D25, 14K07

Retrieve articles in all journals with MSC: 11D25, 14K07


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0717717-X
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society