Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Integer points on $y^{2}=x^{3}-7x+10$
HTML articles powered by AMS MathViewer

by Andrew Bremner and Nicholas Tzanakis PDF
Math. Comp. 41 (1983), 731-741 Request permission

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
    W. E. H. Berwick, "Algebraic number fields with two independent units," Proc. London Math. Soc. v. 34, 1932, pp. 360-378. G. Billing, "Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlecht eins," Nova Acta Reg. Soc. Scient. Upsaliensis, Ser. IV, v. 11, 1938.
  • B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. MR 0376533
  • B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25. MR 146143, DOI 10.1515/crll.1963.212.7
  • Arne J. Brentjes, A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields, J. Reine Angew. Math. 326 (1981), 18–44. MR 622343, DOI 10.1515/crll.1981.326.18
  • J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193
  • A. O. L. Atkin and B. J. Birch (eds.), Computers in number theory, Academic Press, London-New York, 1971. MR 0314733
  • B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
  • Raphael Finkelstein and Hymie London, On Mordell’s equation $y^{2}-k=x^{3}$: An interesting case of Sierpiński, J. Number Theory 2 (1970), 310–321. MR 268120, DOI 10.1016/0022-314X(70)90058-2
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
  • Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817, DOI 10.1007/978-3-662-07010-9
  • Serge Lang, Conjectured Diophantine estimates on elliptic curves, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171. MR 717593
  • W. Ljunggren, On the diophantine equation $y^{2}-k=x^{3}$, Acta Arith. 8 (1962/63), 451–463. MR 158859, DOI 10.4064/aa-8-4-451-463
  • Jean-François Mestre, Construction d’une courbe elliptique de rang $\geq 12$, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 12, 643–644 (French, with English summary). MR 688896
  • L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
  • 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.
  • Th. Skolem, The use of a $p$-adic method in the theory of diophantine equations, Bull. Soc. Math. Belg. 1954 (1954), 83–95 (1955). MR 72155
  • Th. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen, 8de Skand. mat. Kongr., Stockholm, 1934.
  • J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
  • N. Tzanakis, "The Diophantine equation ${x^3} - 3x{y^2} - {y^3} = 1$ and related equations," J. Number Theory. (To appear.)
  • A. Wiman, Über die Punkte mit ganzzahligen Koordinaten auf gewissen Kurven dritter Ordnung, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Institution, Lund, 1954, pp. 317–323 (German). MR 0065585
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 11D25, 14K07
  • Retrieve articles in all journals with MSC: 11D25, 14K07
Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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