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Complete solutions to families of quartic Thue equations


Author: Attila Pethő
Journal: Math. Comp. 57 (1991), 777-798
MSC: Primary 11D25
DOI: https://doi.org/10.1090/S0025-5718-1991-1094956-7
MathSciNet review: 1094956
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Abstract: Using a method due to E. Thomas, we prove that if $ \vert a\vert > 9.9 \cdot {10^{27}}$ then the Diophantine equations

$\displaystyle {x^4} - a{x^3}y - {x^2}{y^2} + ax{y^3} + {y^4} = 1$

and

$\displaystyle {x^4} - a{x^3}y - 3{x^2}{y^2} + ax{y^3} + {y^4} = \pm 1$

have exactly twelve solutions, namely $ (x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \mp 1, \pm 1), ( \pm a, \pm 1), ( \pm 1, \mp a)$ and eight solutions, $ (x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \pm 1, \mp 1)$ , respectively.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1094956-7
Keywords: Thue equation, linear forms in the logarithms of algebraic numbers
Article copyright: © Copyright 1991 American Mathematical Society

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