On the diophantine equation \(| ax^n-by^n|=1\). (English) Zbl 0892.11041
In this paper the authors show that the equation of the title with \(b \geq a\) and \(n\geq 3\) has at most one integer solution \((x,y)\in\mathbb{N}\times\mathbb{N}\) with the possible exceptions of values of \(a, b\) and \(n\) such that
\[
b=a+1,\quad 2\leq a\leq \min\{0.3 n,83\},\quad 17 \leq n \leq 347.
\]
This is achieved in a novel way by combining results on linear forms in logarithms with results on Padé approximations to hypergeometric functions. All the relevant computations are explained in great detail, including all the necessary number field data such as the fundamental units and various ideal generators. This data is required for a variety of number fields of degree up to \(13\). Four different techniques for reducing the upper bounds, derived from estimates on linear forms in logarithms, are given and contrasted. These are the classical method of Tzanakis and de Weger, using simultaneous linear forms, the method of Bilu and Hanrot using two-dimensional lattices and a new method which is a combination of the others.
Reviewer: Nigel Smart (Bristol)
MSC:
| 11Y50 | Computer solution of Diophantine equations |
| 11D41 | Higher degree equations; Fermat’s equation |
References:
| [1] | A. Baker, Rational approximations to certain algebraic numbers, Proc. London Math. Soc. (3) 14 (1964), 385 – 398. · Zbl 0131.29102 · doi:10.1112/plms/s3-14.3.385 |
| [2] | A. Baker, Rational approximations to \root3\of2 and other algebraic numbers, Quart. J. Math. Oxford Ser. (2) 15 (1964), 375 – 383. · Zbl 0222.10036 · doi:10.1093/qmath/15.1.375 |
| [3] | 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 (1967/1968), 173 – 191. · Zbl 0157.09702 · doi:10.1098/rsta.1968.0010 |
| [4] | A. Baker and H. Davenport, The equations 3\?²-2=\?² and 8\?²-7=\?², Quart. J. Math. Oxford Ser. (2) 20 (1969), 129 – 137. · Zbl 0177.06802 · doi:10.1093/qmath/20.1.129 |
| [5] | A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19 – 62. · Zbl 0788.11026 · doi:10.1515/crll.1993.442.19 |
| [6] | M. Bennett, “Effective measures of irrationality for certain algebraic numbers”, J. Austral. Math. Soc., to appear. · Zbl 0880.11055 |
| [7] | M. Bennett, “Explicit lower bounds for rational approximation to algebraic numbers”, Proc. London Math. Soc., to appear. · Zbl 0879.11038 |
| [8] | Yu. Bilu and G. Hanrot, “Solving Thue equations of high degree”, J. Number Theory 60 [1996], 373-392. CMP 97:02 · Zbl 0867.11017 |
| [9] | G. V. Chudnovsky, On the method of Thue-Siegel, Ann. of Math. (2) 117 (1983), no. 2, 325 – 382. · Zbl 0518.10038 · doi:10.2307/2007080 |
| [10] | H. Cohen, Algorithmic Algebraic Number Theory, Springer Verlag, Berlin, 1993. |
| [11] | H. Darmon and L. Merel, “Winding quotients and some variants of Fermat’s Last Theorem”, J. Reine Angew. Math., to appear. · Zbl 0976.11017 |
| [12] | B.N. Delone, “Solution of the indeterminate equation \( X^3 q + Y^3 = 1 \)”, Izv. Akad. Nauk SSR (6) 16 [1922], 253-272. |
| [13] | B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. · Zbl 0133.30202 |
| [14] | Y. Domar, “On the Diophantine equation \( | A x^n - B y^n | = 1, n \geq 5 \)”, Math. Scand. 2 [1954], 29-32. · Zbl 0056.03602 |
| [15] | J.-H. Evertse, Upper Bounds for the Numbers of Solutions of Diophantine Equations, PhD Thesis, Leiden, 1983. · Zbl 0517.10016 |
| [16] | Jan-Hendrik Evertse, On the equation \?\?\(^{n}\)-\?\?\(^{n}\)=\?, Compositio Math. 47 (1982), no. 3, 289 – 315. · Zbl 0498.10014 |
| [17] | J.-H. Evertse, On the representation of integers by binary cubic forms of positive discriminant, Invent. Math. 73 (1983), no. 1, 117 – 138. , https://doi.org/10.1007/BF01393828 J.-H. Evertse, Erratum: ”On the representation of integers by binary cubic forms of positive discriminant”, Invent. Math. 75 (1984), no. 2, 379. · Zbl 0526.10012 · doi:10.1007/BF01388571 |
| [18] | S. Hyrrö, “Über die Gleichung \( a x^n - b y^n = z \) und das Catalansche Problem”, Ann. Acad. Sci. Fenn. Ser. A I, No. 355, 1964, 50 pp. |
| [19] | Michel Laurent, Maurice Mignotte, and Yuri Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation, J. Number Theory 55 (1995), no. 2, 285 – 321 (French, with English summary). · Zbl 0843.11036 · doi:10.1006/jnth.1995.1141 |
| [20] | A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515 – 534. · Zbl 0488.12001 · doi:10.1007/BF01457454 |
| [21] | W.J. LeVeque, Topics in Number Theory, Addison-Wesley, Reading, Mass. 1956. · Zbl 0070.03804 |
| [22] | W. Ljunggren, “Einige Eigenschaften der Einheitenreeller quadratischer und rein biquadratischer Zahlkörper mit Anwendung auf die Lösung einer Klasse von bestimmter Gleichungen vierten Grades”, Det Norske Vidensk. Akad. Oslo Skrifter I, No. 12 [1936], 1-73. |
| [23] | W. Ljunggren, “On an improvement of a theorem of T. Nagell concerning the Diophantine equation \( A x^3 + B y^3 = C \)”, Math. Scan. 1 [1953], 297-309. · Zbl 0051.27803 |
| [24] | Maurice Mignotte, A note on the equation \?\?\(^{n}\)-\?\?\(^{n}\)=\?, Acta Arith. 75 (1996), no. 3, 287 – 295. · Zbl 0861.11025 |
| [25] | Maurice Mignotte and Benjamin M. M. de Weger, On the Diophantine equations \?²+74=\?\(^{5}\) and \?²+86=\?\(^{5}\), Glasgow Math. J. 38 (1996), no. 1, 77 – 85. · Zbl 0847.11011 · doi:10.1017/S0017089500031293 |
| [26] | L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. · Zbl 0188.34503 |
| [27] | Julia Mueller, Counting solutions of |\?\?^{\?}-\?\?^{\?}|\le \?, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 503 – 513. · Zbl 0632.10014 · doi:10.1093/qmath/38.4.503 |
| [28] | J. Mueller and W. M. Schmidt, Thue’s equation and a conjecture of Siegel, Acta Math. 160 (1988), no. 3-4, 207 – 247. · Zbl 0655.10016 · doi:10.1007/BF02392276 |
| [29] | T. Nagell, “Solution complète de quelques équations cubiques à deux indéterminées”, J. de Math. (9) 4 [1925], 209-270. |
| [30] | Paulo Ribenboim, Catalan’s conjecture, Academic Press, Inc., Boston, MA, 1994. Are 8 and 9 the only consecutive powers?. · Zbl 0824.11010 |
| [31] | T. N. Shorey, Perfect powers in values of certain polynomials at integer points, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 195 – 207. · Zbl 0598.10029 · doi:10.1017/S0305004100064112 |
| [32] | T. N. Shorey and R. Tijdeman, New applications of Diophantine approximations to Diophantine equations, Math. Scand. 39 (1976), no. 1, 5 – 18. · Zbl 0341.10017 · doi:10.7146/math.scand.a-11641 |
| [33] | C.L. Siegel, “Die Gleichung \( a x^n - b y^n = c \)”, Math. Ann. 144 [1937], 57-68. · JFM 63.0117.01 |
| [34] | V. Tartakovski[??]i, “Auflösung der Gleichung \( x^4 - \rho y^4 = 1 \)”, Izv. Akad. Nauk SSR (6) 20 [1926], 301-324. |
| [35] | A. Thue, “Über Annäherungenswerte algebraischen Zahlen”, J. Reine Angew. Math. 135 [1909], 284-305. |
| [36] | R. Tijdeman, Applications of the Gel\(^{\prime}\)fond-Baker method to rational number theory, Topics in number theory (Proc. Colloq., Debrecen, 1974) North-Holland, Amsterdam, 1976, pp. 399 – 416. Colloq. Math. Soc. János Bolyai, Vol. 13. |
| [37] | N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), no. 2, 99 – 132. · Zbl 0657.10014 · doi:10.1016/0022-314X(89)90014-0 |
| [38] | B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. · Zbl 0687.10013 |
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