Diophantine approximation of ternary linear forms

Author:
T. W. Cusick

Journal:
Math. Comp. **25** (1971), 163-180

MSC:
Primary 10F99

DOI:
https://doi.org/10.1090/S0025-5718-1971-0296022-4

MathSciNet review:
0296022

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Abstract: The paper gives an efficient method for finding arbitrarily many solutions in integers *x, y, z* of the Diophantine inequality , where defines a totally real cubic field *F* over the rationals, the numbers 1, 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.

**[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.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0296022-4

Keywords:
Diophantine inequality,
ternary linear forms,
totally real cubic field

Article copyright:
© Copyright 1971
American Mathematical Society