On quartic Thue equations with trivial solutions
Author:
R. J. Stroeker
Journal:
Math. Comp. 52 (1989), 175187
MSC:
Primary 11D25
MathSciNet review:
946605
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Abstract: Let K be a quartic number field with negative absolute discriminant and let be its real quadratic subfield, with . Moreover, assume K to be embedded in the reals. Further, let generate the subgroup of units relative to L in the group of positive units of K. Under certain conditions, which can be explicitly checked, and for suitable linear forms and with coefficients in , the diophantine equation which is a quartic Thue equation in the indeterminates u and v, has only trivial solutions, that is, solutions given by . Information on a substantial number of equations of this type and their associated number fields is incorporated in a few tables.
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 A. Baker, "Contributions to the theory of diophantine equations I: On the representation of integers by binary forms, II: The diophantine equation ," Philos. Trans. Roy. Soc. London. Ser A, v. 263, 1968, pp. 173208. MR 0228424 (37:4005)
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 A. Baker & H. Davenport, "The equations and ," Quart. J. Math. Oxford Ser. (2), v. 20, 1969, pp. 129137. MR 0248079 (40:1333)
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 A. Bremner, "A diophantine equation arising from tight 4designs," Osaka J. Math., v. 16, 1979, pp. 353356. MR 539591 (80j:10026)
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 L. Holtzer, Zahlentheorie, Teil I, Math. Naturw. Bibl. 13, Teubner Verlag, Leipzig, 1958.
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 H. London & R. Finkelstein, "On Mordell's equation ," Bowling Green State Univ. Press, 1973. MR 0340172 (49:4928)
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 L. J. Mordell, Diophantine Equations, Pure and Appl. Math., vol. 30, Academic Press, New York, 1969. MR 0249355 (40:2600)
 [7]
 Trygve Nagell, "Sur quelques questions dans la théorie des corps biquadratiques," Ark. Mat., v. 4, 1962, pp. 347376. MR 0150124 (27:127)
 [8]
 Ken Nakamula, "Class number calculation and elliptic unit II," Proc. Japan Acad., v. 57A, 1981, pp. 117120. MR 605295 (82k:12007b)
 [9]
 Ken Nakamula, "Class number calculation of a quartic field from the elliptic unit," Acta Arith., v. 45, 1985, pp. 215227. MR 808022 (87a:11109)
 [10]
 R. J. Stroeker, "On the diophantine equation in connection with the existence of nontrivial tight 4designs," Indag. Math., v. 43, 1981, pp. 353358. MR 632174 (82j:10032)
 [11]
 R. J. Stroeker, "On classes of biquadratic diophantine equations with trivial solutions only," Abstracts Amer. Math. Soc., v. 35, 1984, p. 441.
 [12]
 R. J. Stroeker & R. Tijdeman, "Diophantine equations," in Computational Methods in Number Theory, Part II, MC Tracts 155, Centre Math. Comp. Sci., Amsterdam, 1982, pp. 321369. MR 702521 (84i:10014)
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 R. J. Stroeker & N. Tzanakis, "On the application of Skolem's padic method to the solution of Thue equations," J. Number Theory, v. 29, 1988, pp. 166195. MR 945593 (89f:11044)
 [14]
 A. Thue, "Über Annäherungswerte algebraischer Zahlen," J. Reine Angew. Math., v. 135, 1909, pp. 284305.
 [15]
 Nikos Tzanakis, "On the diophantine equation ," Acta Arith., v. 46, 1986, pp. 257269.
 [16]
 N. Tzanakis & B. M. M. de Weger, On the Practical Solution of the Thue Equation, Memorandum 668, Fac. Appl. Math. Un. of Twente, 1987.
 [17]
 Saburô Uchiyama, "Solution of a diophantine problem," Tsukuba J. Math., v. 8, 1984, pp. 131157. MR 747452 (86i:11010)
 [18]
 Theresa P. Vaughan, "The discriminant of a quadratic extension of an algebraic field," Math. Comp., v. 40, 1983, pp. 685707. MR 689482 (84e:12006)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909466054
PII:
S 00255718(1989)09466054
Article copyright:
© Copyright 1989
American Mathematical Society
