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On the practical solution of the Thue equation. (English) Zbl 0657.10014
This article is a break-through. - Every one working in the field of diophantine equations has the feeling that if there are integer solutions these must be “small” (and perhaps in this context “small” should be interpreted as in the range of todays or next two generations computers).
In the case of the equation \(f(X,Y)=m\) (f\(\in {\mathbb{Z}}[X,Y]\) an irreducible binary form of degree at least 3, and m a given nonzero rational integer) Thue has proved that there are only finitely many integer solutions x,y. Baker’s method gives an astronomical high upper bound for these solutions.
The authors consider four cases for Thue’s equation: the “very small”, “small”, “large”, and “very large” ones. They are able, considering the newest work in this field - almost all in preprint form -, to exclude the last (Baker’s) case. They give a very involved reduction algorithm to convert possible “large” solutions into “small” ones (at least logarithmically smaller than the initial ones) to detect that there are no “large” solutions.
A special example is given for reducing a certain upper bound A of the size of \(10^{40}\) by a computational reduction technique based on diophantine approximation theory to \(A\leq 72\). Applying the same method again gives \(A\leq 10.\)
For “small” solutions there is a continued fraction algorithm computing the solutions. “Very small” solutions must be detected by hand.
Implicitly assumed in all these considerations is that the elementary units of the algebraic number field given by the zeros of \(f(X,1)=0\) are known, equally well as the prime factorization of m in this field.
The article concludes with applying the theory of the diophantine equation \(y^ 2=x^ 3-4x+1\) where a complete solution was given by S. P. Mohanty [ibid. 30, No.1, 86-93 (1988; Zbl 0651.10011)] whose proof A. Bremner has claimed to be faulty (there are some implications of type “d \(| ab\) \(\Rightarrow\) d \(| a\) or d \(| b'')\). They arrive from this elliptic curve at a couple of totally real quartic Thue equations which are suited to their method.
This article is in some sense highly technical and the reviewer is not able to check the correctness of all the theorems. Nevertheless reading this article is a much rewarding experience.
Reviewer: B.Richter

11D41 Higher degree equations; Fermat’s equation
11D25 Cubic and quartic Diophantine equations
Full Text: DOI
[1] Agrawal, M.K.; Coates, J.H.; Hunt, D.C.; Van Der Poorten, A.J., Elliptic curves of conductor 11, Math. comp, 35, 991-1002, (1980) · Zbl 0475.14031
[2] Baker, A., Contributions to the theory of Diophantine equations. I. on the representation of integers by binary forms, Philos. trans. roy. soc. London ser. A, 263, 173-191, (1968) · Zbl 0157.09702
[3] Berwick, W.E.H., Algebraic number-fields with two independent units, (), 360-378 · Zbl 0005.34203
[4] Billevicˇ, K.K., On the units of algebraic fields of third and fourth degree, Mat. sb., 40, 82, 123-137, (1956), [Russian]
[5] Billevicˇ, K.K., A theorem on the units of algebraic fields of nth degree, Mat. sb., 64, 106, 145-152, (1964), [Russian]
[6] Blass, J.; Glass, A.M.W.; Meronk, D.B.; Steiner, R.P., Practical solutions to thue equations over the rational integers, (1987), Bowling Green State University, preprint
[7] Borevich, Z.I.; Shafarevich, I.R., ()
[8] Buchmann, J.; Buchmann, J., A generalization of Voronoi’s unit algorithm II, J. number theory, J. number theory, 10, 192-209, (1986)
[9] {\scJ. Buchmann}, The generalized Voronoi algorithm in totally real algebraic number fields, to appear. · Zbl 0586.12004
[10] Delone, B.N.; Faddeev, D.K., (), Transl. of Mathematical Monographs · Zbl 0061.09001
[11] Dickson, L.E., ()
[12] Ellison, W.J., (), Lab. Th. Nombr. CNRS, Exp. 11
[13] Ellison, W.J.; Ellison, F.; Pesek, J.; Stahl, C.E.; Stall, D.S., The Diophantine equation y2 + k = x3, J. number theory, 4, 107-117, (1972) · Zbl 0236.10010
[14] Fincke, U.; Pohst, M., Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. comp., 44, 463-471, (1985) · Zbl 0556.10022
[15] Hardy, G.H.; Wright, E.M., ()
[16] Lenstra, A.K.; Lenstra, H.W.; Lovász, L., Factoring polynomials with rational coefficients, Math. ann., 261, 515-534, (1982) · Zbl 0488.12001
[17] Mohanty, S.P., Integer points of y2 = x3 − 4x + 1, J. number theory, 30, 86-93, (1988) · Zbl 0651.10011
[18] Mordell, L.J., ()
[19] {\scA. Pethöand R. Schulenberg}, Effektives Lösen von Thue Gleichungen, Publ. Math. Debrecen, in press.
[20] Pethö, A.; De Weger, B.M.M., Products of prime powers in binary recurrence sequences. I. the hyperbolic case, with an application to the generalized Ramanujan-nagell equation, Math. comp., 47, 713-727, (1986) · Zbl 0623.10011
[21] Pohst, M.; Zassenhaus, H.; Pohst, M.; Zassenhaus, H., On effective computation of fundamental units II, Math. comp., Math. comp., 38, 293-329, (1982) · Zbl 0493.12005
[22] Shorey, T.N.; Tijdeman, R., ()
[23] Sprindzˇuk, V.G., (), [Russian]
[24] Steiner, R.P., On Mordell’s equation: A problem of Stolarsky, Math. comp., 46, 703-714, (1986) · Zbl 0601.10011
[25] Thue, A., Über annäherungswerte algebraischer zahlen, J. reine angew. math., 135, 284-305, (1909) · JFM 40.0265.01
[26] Tzanakis, N., On the Diophantine equation x2 − dy4 = k, Acta arith., 46, 257-269, (1986)
[27] Tzanakis, N., On the practical solution of the thue equation, an outline, (), in press · Zbl 0703.11013
[28] Tzanakis, N.; De Weger, B.M.M., (), Memorandum No. 668
[29] {\scN. Tzanakis and B. M. M. De Weger}, On the practical solution of the Thue-Mahler equation, in preparation. · Zbl 0738.11030
[30] Wadschmidt, M., A lower bound for linear forms in logarithms, Acta arith, 37, 257-283, (1980)
[31] De Weger, B.M.M., Solving exponential Diophantine equations using lattice basis reduction algorithms, J. number theory, 26, 325-367, (1987) · Zbl 0625.10013
[32] De Weger, B.M.M.; De Weger, B.M.M., Algorithms for Diophantine equations, (), Also to appear as a · Zbl 0687.10013
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