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An algorithm to determine the points with integral coordinates in certain elliptic curves. (English) Zbl 0923.11036

An algorithm is described, that determines the terms of a Lucas sequence of the first kind which are of the form \(k\)-times a square. Some applications of this algorithm are discussed. The paper is based on previous ones [e.g., P. Ribenboim, Port. Math. 46, 159-175 (1989; Zbl 0687.10005) and J. Sichuan Univ., Nat. Sci. Ed. 26, Spec. Issue, 196-200 (1989; Zbl 0709.11014)].

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11G05 Elliptic curves over global fields
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References:

[1] Bumby, R. T., The diophantine equation \(3x^2y^2\), Math. Scand., 21, 141-148 (1967) · Zbl 0169.37402
[2] Cohn, J. H.E., On square Fibonacci numbers, J. London Math. Soc., 39, 537-540 (1964) · Zbl 0127.26705
[3] Cohn, J. H.E., Square Fibonacci numbers, etc, Fibonacci Quart., 2, 109-113 (1964) · Zbl 0126.07201
[4] Cohn, J. H.E., Lucas and Fibonacci numbers and some diophantine equations, Proc. Glasgow Math. Assoc., 7, 24-28 (1965) · Zbl 0127.01902
[5] Cohn, J. H.E., Eight diophantine equations, Proc. London Math. Soc. (3), 16, 153-166 (1966) · Zbl 0136.02806
[6] Cohn, J. H.E., The diophantine equation \(x^2 Dy^4\), J. London Math. Soc., 40, 475-476 (1967) · Zbl 0163.04003
[7] Cohn, J. H.E., Five diophantine equations, Math. Scand., 21, 61-70 (1967) · Zbl 0169.37401
[8] Cohn, J. H.E., Some quartic diophantine equations, Pacific J. Math, 26, 233-243 (1968) · Zbl 0191.04902
[9] Cohn, J. H.E., Squares in some recurrence sequences, Pacific J. Math., 41, 631-646 (1972) · Zbl 0248.10016
[10] Ljunggren, W., Einige Eigenschaften der Einheiten reeller quadratischer und rein biquadratisher Zahlkörper mit Anwendungen auf die Lösung einer Klasse unbestimmter Gleichungen vierten Grades, Skr. Norske Vidensk. Akad. Oslo I Klasse, 12 (1936) · Zbl 0016.00802
[11] Ljunggren, W., Über die unbestimmte Gleichung \(Ax^2 By^4C\), Arch. Math.-Naturvid., 41, 3-18 (1938) · JFM 64.0975.05
[12] Ljunggren, W., Zur Theorie der Gleichung \(x^2 Dy^4\), Abh. Norsk Vid. Akad. Oslo, 1, 1-27 (1942) · JFM 68.0068.01
[13] Ljunggren, W., New propositions about the indeterminate equation\(x^n\)−\(1/x y^q \), Norske Mat. Tidsskrift, 25, 17-20 (1943)
[14] Ljunggren, W., On the diophantine equation \(x^2 Ay^4\), Kong. Norske Vidensk. Selskab Vorhandl., 21, 82-84 (1951) · Zbl 0046.26503
[15] Ljunggren, W., Some remarks on the diophantine equations \(x^2 Dy^4x^4 Dy^2\), J. London Math. Soc., 41, 542-544 (1966) · Zbl 0147.02505
[16] McDaniel, W. L.; Ribenboim, P., The square terms in Lucas sequences, J. Number Theory, 58, 104-123 (1996) · Zbl 0851.11011
[17] Mignotte, M.; Pethö, A., Sur les carrés dans certaines suites de Lucas, J. Th. Nombres Bordeaux, 5, 333-341 (1993) · Zbl 0795.11007
[18] Pethö, A., Perfect powers in second order linear recurrences, J. Number Theory, 15, 5-13 (1982) · Zbl 0488.10009
[19] Ribenboim, P., The Fibonnacci numbers and the Arctic Ocean, (Behara, M.; Fritsch, R.; Lintz, R. G., Symposia Gaussian a Conf. A. (1995), de Gruyter: de Gruyter Berlin), 41-83 · Zbl 0858.11011
[20] Ribenboim, P., Square-classes of Fibonacci numbers, Portugal Math., 46, 159-175 (1989) · Zbl 0687.10005
[21] Ribenboim, P., Square-classes of\(a^n\)−\(1/a a^n +1\), J. Sichuan Univ. Nat. Sci. Ed., 26, 196-199 (1989)
[22] Ribenboim, P.; McDaniel, W. L., Square-classes in Lucas sequences, Portugal. Math., 48, 469-473 (1991) · Zbl 0760.11005
[23] Ribenboim, P.; McDaniel, W. L., The square-classes of Lucas sequences having odd parameters, C.R. Math. Rep. Acad. Sci. Canada, 18, 223-226 (1996) · Zbl 0882.11011
[24] Shorey, T. N.; Stewart, C. L., On the diophantine equation \(ax^{2t} + bx^ty cy^2\), Math. Scand., 52, 24-36 (1983) · Zbl 0491.10016
[25] Wyler, O., Squares in the Fibonacci series, Amer. Math. Monthly, 71, 220-222 (1964)
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