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.


On the resolution of inhomogeneous norm form equations in two dominating variables
HTML articles powered by AMS MathViewer

by István Gaál PDF
Math. Comp. 51 (1988), 359-373 Request permission


Applying Baker’s well-known method and the reduction procedure described by Baker and Davenport, we give a numerical algorithm for finding all solutions of inhomogeneous Thue equations of type \[ {N_{K/Q}}(x + \alpha y + \lambda ) = 1\] in the variables $x,y \in Z$ and $\lambda \in {Z_K}$ with $\lceil \lambda \rceil < (\max |x|,|y|){)^{1/2}}$, where $K = Q(\alpha )$ is a totally real cubic field.
  • A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser. A 263 (1967/68), 173–191. MR 228424, DOI 10.1098/rsta.1968.0010
  • Alan Baker, Transcendental number theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. MR 1074572
  • A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
  • J. Coates, An effective $p$-adic analogue of a theorem of Thue, Acta Arith. 15 (1968/69), 279–305. MR 242768, DOI 10.4064/aa-15-3-279-305
  • W. J. Ellison, On sums of squares in $Q^{1/2}(X)$ etc, Séminaire de Théorie des Nombres, 1970-1971 (Univ. Bordeaux I, Talence), Exp. No. 9, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1971, pp. 5. MR 0392880
  • W. J. Ellison, F. Ellison, J. Pesek, C. E. Stahl, and D. S. Stall, The Diophantine equation $y^{2}+k=x^{3}$, J. Number Theory 4 (1972), 107–117. MR 316376, DOI 10.1016/0022-314X(72)90058-3
  • I. Gaál, Norm form equations with several dominating variables and explicit lower bounds for inhomogeneous linear forms with algebraic coefficients, Studia Sci. Math. Hungar. 19 (1984), no. 2-4, 399–411. MR 874508
  • I. Gaál, Norm form equations with several dominating variables and explicit lower bounds for inhomogeneous linear forms with algebraic coefficients, Studia Sci. Math. Hungar. 19 (1984), no. 2-4, 399–411. MR 874508
  • K. Győry, Sur certaines généralisations de l’équation de Thue-Mahler, Enseign. Math. (2) 26 (1980), no. 3-4, 247–255 (1981) (French). MR 610525
  • K. Györy, Résultats Effectifs sur la Représentation des Entiers par des Formes Décomposables, Queen’s Papers in Pure and Appl. Math., No. 56, Kingston, Ontario, Canada, 1980.
  • K. Győry, On $S$-integral solutions of norm form, discriminant form and index form equations, Studia Sci. Math. Hungar. 16 (1981), no. 1-2, 149–161. MR 703653
  • K. Győry, Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains, Acta Math. Hungar. 42 (1983), no. 1-2, 45–80. MR 716553, DOI 10.1007/BF01960551
  • K. Győry and Z. Z. Papp, Norm form equations and explicit lower bounds for linear forms with algebraic coefficients, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 245–257. MR 820227
  • Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456
  • S. V. Kotov, On Diophantine Equations of Norm Form Type II, Inst. Mat. Akad. Nauk BSSR, Preprint No. 10, Minsk, 1980. (Russian) S. V. Kotov, Effective Bounds for Linear Forms with Algebraic Coefficients in Archimedean and p-Adic Metrics, Inst. Mat. Akad. Nauk BSSR, Preprint No. 24, Minsk, 1981. (Russian)
  • S. V. Kotov, An effective estimate for the bounds on the number of solutions of a class of Diophantine norm form equations, Mat. Zametki 33 (1983), no. 6, 801–806 (Russian). MR 709218
  • Attila Pethő, On the resolution of Thue inequalities, J. Symbolic Comput. 4 (1987), no. 1, 103–109. MR 908418, DOI 10.1016/S0747-7171(87)80059-7
  • A. Pethő and R. Schulenberg, Effektives Lösen von Thue Gleichungen, Publ. Math. Debrecen 34 (1987), no. 3-4, 189–196 (German). MR 934900
  • T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR 891406, DOI 10.1017/CBO9780511566042
  • V. G. Sprindžuk, Representation of numbers by the norm forms with two dominating variables, J. Number Theory 6 (1974), 481–486. MR 354567, DOI 10.1016/0022-314X(74)90044-4
  • Ray P. Steiner, On Mordell’s equation $y^2-k=x^3$: a problem of Stolarsky, Math. Comp. 46 (1986), no. 174, 703–714. MR 829640, DOI 10.1090/S0025-5718-1986-0829640-3
  • R. Tijdeman, Hilbert’s seventh problem: on the Gel′fond-Baker method and its applications, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 241–268. MR 0434974
  • Michel Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257–283. MR 598881, DOI 10.4064/aa-37-1-257-283
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 11D57
  • Retrieve articles in all journals with MSC: 11D57
Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Math. Comp. 51 (1988), 359-373
  • MSC: Primary 11D57
  • DOI:
  • MathSciNet review: 942162