Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the solutions of a family
of quartic Thue equations

Author: Alain Togbé
Journal: Math. Comp. 69 (2000), 839-849
MSC (1991): Primary 11D25, 11D72, 11D85, 11J86, 11R16, 11Y50
Published electronically: May 17, 1999
MathSciNet review: 1648411
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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: $(1,0),\; (-1,0),\; (0,1),\; (0,-1)$.

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

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 11D25, 11D72, 11D85, 11J86, 11R16, 11Y50

Retrieve articles in all journals with MSC (1991): 11D25, 11D72, 11D85, 11J86, 11R16, 11Y50

Additional Information

Alain Togbé
Affiliation: Département de Mathematiques et de Statistique, Université Laval, Québec, Québec, G1K 7P4 Canada

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
Published electronically: May 17, 1999
Article copyright: © Copyright 2000 American Mathematical Society