Evertse, J.-H. On equations in S-units and the Thue-Mahler equation. (English) Zbl 0521.10015 Invent. Math. 75, 561-584 (1984). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 14 ReviewsCited in 56 Documents MSC: 11D57 Multiplicative and norm form equations 11D61 Exponential Diophantine equations 11R27 Units and factorization 11D88 \(p\)-adic and power series fields Keywords:equations in S-units; Thue-Mahler equation; Archimedean valuations; binary form of degree n; generalisation of Ramanujan-Nagell equation; height functions Citations:Zbl 0102.036 PDF BibTeX XML Cite \textit{J. H. Evertse}, Invent. Math. 75, 561--584 (1984; Zbl 0521.10015) Full Text: DOI EuDML OpenURL References: [1] Beukers, F.: The generalised Ramanujan-Nagell equation. Thesis, Leiden 1979 · Zbl 0412.10009 [2] Beukers, F.: On the generalized Ramanujan-Nagell equation. I. Acta Arith.38, 389-410 (1980/81) [3] Beukers, F.: On the generalized Ramanujan-Nagell equation. II. Acta Arith.39, 113-123 (1981) · Zbl 0377.10012 [4] Choodnovsky, G.V.: The Gel’fond-Baker method in problems of diophantine approximation. Coll. Math. Soc. János Bolyai13, 19-30 (1974) [5] Delone, B.N.: Über die Darstellung der Zahlen durch die binären kubischen Formen von negativer Diskriminante. Math. Z.31, 1-26 (1930) · JFM 55.0722.02 [6] Evertse, J.H.: On the equationax n ?by n =c. Compositio Math.47, 289-315 (1982) · Zbl 0498.10014 [7] Evertse, J.H.: On the representation of integers by binary cubic forms of positive discriminant. Invent. Math.73, 117-138 (1983) · Zbl 0506.10013 [8] Evertse, J.H.: Upper bounds for the numbers of solutions of diophantine equations. MC-tract, Mathematisch Centrum, Amsterdam 1983 · Zbl 0517.10016 [9] Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math.73, 349-366 (1983) · Zbl 0588.14026 [10] Györy, K.: On the number of solutions of linear equations in units of an algebraic number field. Comm. Math. Helv.54, 583-600 (1979) · Zbl 0437.12004 [11] Lewis, D.J., Mahler, K.: Representation of integers by binary forms. Acta Arith.6, 333-363 (1960/61) [12] Mahler, K.: Zur Approximation algebraischer Zahlen. I. Über den größten Primteiler binärer Formen. Math. Ann.107, 691-730 (1933) · Zbl 0006.10502 [13] Mahler, K.: Zur Approximation algebraischer Zahlen. II. Über die Anzahl der Darstellungen ganzer Zahlen durch Binärformen. Math. Ann.108, 37-55 (1933) · Zbl 0006.15604 [14] Mahler, K.: On Thue’s theorem. Austral. Nat. Un. Math. Res. Rep.24, (1982); Math. Scand. in press (1984) · Zbl 0544.10014 [15] Mordell, L.J.: Diophantine equations. London: Academic Press 1969 · Zbl 0188.34503 [16] Nagell, T.: Darstellung ganzer Zahlen durch binäre kubische Formen mit negativer Diskriminante. Math. Z.28, 10-29 (1928) · JFM 54.0174.02 [17] Nagell, T.: The diophantine equationx 2+7=2 n , Norsk Mat. Tidsskr.30, 62-64 (1948); Ark. Mat.4, 185-187 (1960) [18] Parry, C.J.: Thep-adic generalisation of the Thue-Siegel theorem. Acta Math.83, 1-99 (1950) · Zbl 0039.27501 [19] Ramanujan, S.: Collected papers: p. 327. Chelsea Publ. Co., New York (1962) [20] Siegel, C.L.: Die Gleichungax n ?by n =c. Math. Ann.114, 57-68 (1937); Gesammelte Abhandlungen, vol. 2. pp. 8-19. Berlin-Heidelberg-New York: Springer 1966 · Zbl 0015.38902 [21] Silverman, J.H.: Integer points and the rank of Thue elliptic curves. Invent. Math.66, 395-404 (1982) · Zbl 0494.14008 [22] Silverman, J.H.: The Thue equation and height functions. In: Bertrand, D., Waldschmidt, M. eds. Approximations diophantienne et nombres transcendents. Coll. Luminy 1982, pp. 259-270. Boston-Basel-Stuttgart: Birkhäuser 1983 [23] Silverman, J.H.: Quantitative results in diophantine geometry. Preprint, Massachusetts Inst. of Techn. [24] Tartakovskii, V.A.: A uniform estimate of the number of representations of unity by a binary form of degreen?3. Dokl. Akad. Nauk. SSSR193: (1970) (Russian): Soviet Math. Dokl.11, 1026-1027 (1970) · Zbl 0256.12005 [25] Thue, A.: Berechnung aller Lösungen gewisser Gleichungen von der Formax r ?by r =f, Vid. Selsk. Skrifter I. mat-naturv. Kl., Christiania 1918, Nr. 4; Selected Mathematical Papers of Axel Thue, pp. 565-572. Oslo, Bergen, Tromsø (1977) 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.