×

zbMATH — the first resource for mathematics

On the application of Skolem’s p-adic method to the solution of Thue equations. (English) Zbl 0674.10012
The authors show by rather ingenious methods that the only integer solutions of the diophantine equation \[ (*)\quad x^ 4-4x^ 2y^ 2+y^ 4=-47 \] are \((x,y)=(\pm 2,\pm 3)\) and \((\pm 3,\pm 2)\). This seems to be the first case where a totally real binary quartic has been successfully attempted by methods of algebraic number theory.
The authors use a trick of Ljunggren - working in a number field M of degree 8 over \({\mathbb{Q}}\)- to reduce (*) to a p-adic system to which Skolem’s method is successfully applied with prime \(p=71\). The numerical hardest job is to find all “exceptable” units in M for which task they give a powerful reduction algorithm [see also the second author and B. M. M. de Weger, ibid. 31, No.1, 99-132 (1989; Zbl 0657.10014)].
Reviewer: B.Richter

MSC:
11D25 Cubic and quartic Diophantine equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baker, A, Contributions to the theory of Diophantine equations \( I II\): on the representation of integers by binary forms/the Diophantine equation y2 = x3 + k, Philos. trans. royal soc. London ser. A, 263, 173-208, (1967/1968)
[2] Baker, A.; Davenport, H., The equations 3x2 − 2 = y2 and 8x2 − 7 = z2, Quart. J. math. Oxford ser., 20, 129-137, (1969)
[3] Billevič, K.K., On the unit of algebraic fields of third and fourth degree, Math. sbornik, 40, 123-136, (1956)
[4] {\scJ. Blass}et al., On Mordell’s equation y2 + k = x3, preprint.
[5] Blass, J., Practical solutions to thue equations of degree 4 over the rational integers, (1986), Preliminary Report
[6] Borevich, Z.I.; Shafarevich, R., (), (Chap. 4, Section 6)
[7] Bremner, A., Integral generators in a certain quartic field and related Diophantine equations, Michigan math. J., 32, 295-319, (1985) · Zbl 0585.12005
[8] {\scJ. Buchmann}, On the computation of units and class numbers by a generalization of Lagrange’s algorithm, preprint. · Zbl 0615.12001
[9] Coghlan, F.B.; Stephens, N.M.; Coghlan, F.B.; Stephens, N.M., The Diophantine equation y2 − k = x3, (), 199-205, No. 2 · Zbl 0217.03601
[10] Ellison, W.J., (), Sém. Th. Nombr. 1970-1971, Exp. No. 11
[11] Ellison, W.J., The Diophantine equation y2 + k = x3, J. number theory, 4, 107-117, (1972) · Zbl 0236.10010
[12] Lang, S., Algebra, (1970), Addison-Wesley Reading, MA
[13] Lewis, D.J., Diophantine equations: p-adic methods, () · Zbl 0218.10035
[14] Ljunggren, W., Einige bemerkungen über die darstellung ganzer zahlen durch binäre kubische formen mit positiver diskriminante, Acta math., 75, 1-21, (1942) · JFM 68.0067.04
[15] {\scW. Ljunggren}, “Diophantine Equations: A p-adic Approach,” Lectures given at the University of Nottingham in 1968. Notes prepared by R. R. Laxton.
[16] Ljunggren, W., On the representation of integers by certain binary cubic and biquadratic forms, Acta arithm., XVII, 379-387, (1971) · Zbl 0216.04102
[17] Mordell, L.J., (), Chap. 23
[18] Matiyasevich, Yu.V., Diophantine representation of enumerable predicates, Izv. akad. nauk. SSSR ser. mat., 35, 3-30, (1971) · Zbl 0235.02039
[19] {\scA. Pethö and R. Schulenberg}, Effectives Lösen von Thue Gleichungen, Publ. Math. Debrecen, to appear.
[20] Pohst, M.; Zassenhaus, H.; Pohst, M.; Zassenhaus, H., On effective computation of fundamental units I & II (with Peter weiler), Math. comp., Math. comp., 38, 293-329, (1982) · Zbl 0493.12005
[21] Rudman, R.J.; Steiner, R.P., A generalization of Berwick’s unit algorithm, J. number theory, 10, 16-34, (1978) · Zbl 0372.12010
[22] Skolem, T., Ein verfahren zur behandlung gewisser exponentialer gleichungen, (), 163-188 · Zbl 0011.39201
[23] Skolem, T., The use of p-adic methods in the theory of Diophantine equations, Bull. soc. math. belg., 7, 83-95, (1955) · Zbl 0065.27405
[24] Steiner, R.P., On the units in algebraic number fields, (), 413-435
[25] Steiner, R.P., On Mordell’s equation y2 − k = x3: A problem of Stolarsky, Math. comp., 46, 703-714, (1986) · Zbl 0601.10011
[26] Stroeker, R.J., On the Diophantine equation x3 − dy2 = 1, Nieuw arch. wisk. (3), 24, 231-255, (1976)
[27] Stroeker, R.J.; Tzanakis, N., (), Report 8607/B
[28] Thue, A., Über annäherungswerte algebraischer zahlen, J. reine angew. math., 135, 284-305, (1909) · JFM 40.0265.01
[29] Tzanakis, N., The Diophantine equation x3 − 3xy2 − y3 = 1 and related equations, J. number theory, 18, No. 2, 192-205, (1984), Corrigendum, J. Number Theory, {\bf19} (1984), 296
[30] Tzanakis, N., On the Diophantine equation x2 − dy4 = k, Acta arithm., 46, No. 3, 257-269, (1986)
[31] Tzanakis, N., On the Diophantine equation 2x3 + 1 = py2, Manuscripta math., 54, 145-164, (1985)
[32] {\scN. Tzanakis and B. M. M. de Weger}, On the practical solution of Thue equations, in preparation. · Zbl 0738.11030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.