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Conway, J. H. and Jones, A. J. , Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arith. , 30 (1976), 229–240. MR 54#10141 Jones A. J

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Summary  

We consider the relative Thue equations \[X^3 - t X^2 Y - (t+1) X Y^2 -Y^3=\mu,\] where the parameter \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $t$ \end{document}, the root of unity \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mu$ \end{document} and the solutions \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Y$ \end{document} are integers in the same imaginary quadratic number field. We use Baker's method to find all solutions for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $|t|> 2.88 \cdot 10^{33}$ \end{document}.

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exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969–1018. MR 2007f :11031 Siksek S. Classical and modular approaches

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SHOREY, T. N. and TIJDEMAN, R., Exponential Diophantine equations Cambridge Tracts in Mathematics, 87, Cambridge University Press, Cambridge - New York, 1986. MR 88h :11002

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. 2010 38 773 784 Stanley, R. P. , Linear Diophantine equations and local cohomology, Invent. Math. , 68

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] Chen , J. H. , Voutier , P. 1997 Complete solution of the Diophantine equation X 2 +1= dY 4 and a related family of quartic Thue equations J. Number Theory 62 71 – 99 10.1006/jnth.1997.2018 . [7

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Stanley, R. , Linear Diophantine equations and local cohomology, Invent. Math. , 68 (1982), 175–193. Stanley R. Linear Diophantine equations and local cohomology

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