Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Diophantine approximation of ternary linear forms

Author: T. W. Cusick
Journal: Math. Comp. 25 (1971), 163-180
MSC: Primary 10F99
MathSciNet review: 0296022
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper gives an efficient method for finding arbitrarily many solutions in integers x, y, z of the Diophantine inequality $ \vert x + \alpha y + \beta z\vert\max ({y^2},{z^2}) < c$, where $ \alpha $ defines a totally real cubic field F over the rationals, the numbers 1, $ \alpha,\beta $ form an integral basis for F, and c is a constant which can be calculated in terms of parameters of the method. For certain values of c, the method generates all solutions of the inequality.

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

  • [1] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Math. and Math. Phys., no. 45, Cambridge Univ. Press, New York, 1957. MR 19, 396. MR 0087708 (19:396h)
  • [2] H. Hancock, Development of the Minkowski Geometry of Numbers, Macmillan, New York, 1939, pp. 371-452. MR 1, 67.
  • [3] H. Minkowski, "Zur Theorie der Kettenbrüche," in Gesammelte Abhandlungen. Vol. I, Teubner, Leipzig, 1911, pp. 278-292.
  • [4] L. G. Peck, "Simultaneous rational approximations to algebraic numbers," Bull. Amer. Math. Soc., v. 67, 1961, pp. 197-201. MR 23 #A111. MR 0122772 (23:A111)
  • [5] M. Zeisel, "Zur Minkowskischen Parallelepipedapproximation," S.-B. Akad. Wiss. Wien, v. 126, 1917, pp. 1221-1247.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10F99

Retrieve articles in all journals with MSC: 10F99

Additional Information

Keywords: Diophantine inequality, ternary linear forms, totally real cubic field
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society