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On the solutions of a family of quartic Thue equations

Author(s): Alain Togbé.
Journal: Math. Comp. 69 (2000), 839-849.
MSC (1991): Primary 11D25, 11D72, 11D85, 11J86, 11R16, 11Y50
Posted: May 17, 1999
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Abstract: In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for $n\leq 5\times 10^6$ and $n \geq N=1.191\times 10^{19}$, with $n,\, n+2,\, n^2+4$ square-free, the Thue equation

\begin{displaymath}\Phi _n(x,y)=x^4 - n^2 x^3 y -(n^3+2n^2+4n+2) x^2 y^2 - n^2 x y^3 + y^4 = 1 \end{displaymath}

has no integral solution except the trivial ones: $(1,0),\; (-1,0),\; (0,1),\; (0,-1)$.

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Additional Information:

Alain Togbé
Affiliation: Département de Mathematiques et de Statistique, Université Laval, Québec, Québec, G1K 7P4 Canada
Address at time of publication: Department of Mathematics & Computer Science, Greenville College, 315 E. College Avenue, Greenville, IL 62246
Email: atogbe@mat.ulaval.ca

DOI: 10.1090/S0025-5718-99-01100-X
PII: S 0025-5718(99)01100-X
Keywords: Quartic equations, equations in many variables, representation problems, linear forms in logarithms, Baker's method, quartic extensions, computer solution of Diophantine equations
Received by editor(s): March 3, 1998
Received by editor(s) in revised form: April 28, 1998
Posted: May 17, 1999
Copyright of article: Copyright 2000, American Mathematical Society

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