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2 results

  • A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

  • Yann Bugeaud, Maurice Mignotte, Samir Siksek
  • Journal:
    Canadian Journal of Mathematics / Volume 60 / Issue 3 / 01 June 2008
    Published online by Cambridge University Press:
    20 November 2018, pp. 491-519
    Print publication:
    01 June 2008
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  • We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation

    $${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$

    in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

  • Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

  • Yann Bugeaud, Maurice Mignotte, Samir Siksek
  • Journal:
    Compositio Mathematica / Volume 142 / Issue 1 / January 2006
    Published online by Cambridge University Press:
    13 January 2006, pp. 31-62
    Print publication:
    January 2006
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  • This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue–Nagell equation x2 + D = yn, x, y integers, $n\geq 3$, for D in the range $1 \leq D \leq 100$.