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On the size of the intersection of two Lucas sequences of distinct type II

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Abstract

Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togbé and Walsh proved that the Diophantine equation

$x^2 - a\left( {\frac{{b^k - 1}} {{b - 1}}} \right)^2 = 1$

has at most three solutions in positive integers. Moreover, they showed that if max{a, b } > 4.76 · 1051, then there are at most two positive integer solutions (x, k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.

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References

  1. Baker A, Davenport H. The equations 3x 2 − 2 = y 2 and 8x 2 − 7 = z 2. Quart J Math Oxford, 1969, 20: 129–137

    Article  MATH  MathSciNet  Google Scholar 

  2. Bennett M A, Cipu M, Mignotte M, et al. On the number of solutions of simultaneous Pell equations II. Acta Arith, 2006, 122: 75–122

    Article  MathSciNet  Google Scholar 

  3. Bosma W, Cannon J, Playoust C. The Magma algebra system, I: The user language. J Symbolic Comput, 1997, 24: 235–265

    Article  MATH  MathSciNet  Google Scholar 

  4. Cipu M. Pairs of Pell equations having at most one common solutions in positive integers. An Şt Univ Ovidius Constanţa, 2007, 15: 1–12

    MathSciNet  Google Scholar 

  5. Cipu M, Mignotte M. On the number of solutions to system of Pell equations. J Number Theory, 2007, 125: 356–392

    Article  MATH  MathSciNet  Google Scholar 

  6. He B, Togbé A. Simultaneous Pell equations with single or no solution. Acta Arith, 2008, 133: 369–380

    Google Scholar 

  7. He B, Togbé A, Walsh G P. On the size of the intersection of two Lucas sequences of distinct type. Ann Sci Math Québec, in press

  8. Laurent M. Linear forms in two logarithms and interpolation determinants II. Acta Arith, 2008, 133.4: 325–348

    Article  MathSciNet  Google Scholar 

  9. Li Z G, Xia J Y, Yuan P Z. On some special forms of simultaneous Pell equations. Acta Arith, 2007, 128: 55–67

    Article  MATH  MathSciNet  Google Scholar 

  10. Matveev E M. An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II (in Russian). Izv Ross Akad Nauk Ser Mat, 2000, 64: 125–180. English translation in Izv Math, 2000, 64: 1217–1269

    Google Scholar 

  11. Walsh P G. Sharp bounds for the number of solutions to simultaneous Pellian equations. Acta Arith, 2007, 126: 125–137

    Article  MATH  MathSciNet  Google Scholar 

  12. Walsh P G. A quantitative version of Runge’s theorem on diophantine equations. Acta Arith, 1992, 62: 157–172

    MATH  MathSciNet  Google Scholar 

  13. Yuan P Z. Simultaneous Pell equations. Acta Arith, 2004, 115: 119–132

    Article  MATH  MathSciNet  Google Scholar 

  14. Yuan P Z. On the number of solutions of x 2 − 4m(m+1)y 2 = y 2 − bz 2 = 1. Proc Amer Math Soc, 2004, 132: 1561–1566

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the Romanian Academy, Bucharest, RO-014700, Romania

    Mihai Cipu

  2. U. F. R. de Mathématiques, Université de Strasbourg, Strasbourg, 67084, France

    Maurice Mignotte

  3. Department of Mathematics, Purdue University North Central, Westville, IN, 46391, USA

    Alain Togbé

Authors
  1. Mihai Cipu
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  2. Maurice Mignotte
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  3. Alain Togbé
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Correspondence to Alain Togbé.

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Cipu, M., Mignotte, M. & Togbé, A. On the size of the intersection of two Lucas sequences of distinct type II. Sci. China Math. 54, 1299–1316 (2011). https://doi.org/10.1007/s11425-011-4242-5

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  • DOI: https://doi.org/10.1007/s11425-011-4242-5

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