Borka Jadrijevič: A family of quartic Thue inequalities

This is a joint work with Andrej Dujella.

We prove that the only primitive solutions of the Thue inequality

$\displaystyle \vert x^{4}-4cx^{3}y+(6c+2)x^{2}y^{2}+4cxy^{3}+y^{4}\vert\leq 6c+4,
$

where $ c\geq 4$ is an integer, are $ (x,y)=(\pm 1,0)$, $ (0,\pm 1)$, $ (1,\pm
1) $, $ (-1,\pm 1)$, $ (\pm 1,\mp 2)$, $ (\pm 2,\pm 1).$ Solving Thue equations $ F(x,y)=\mu $ of the special type, using the method of Tzanakis, reduces to solving the system of Pellian equations. The application of Tzanakis method for solving Thue equations has several advantages (see [1,3]). We show that some additional advantages appear when one deals with corresponding Thue inequalities. Namely, the theory of continued fractions can be used in order to determine small values of $ \mu $ for which the equation $ F(x,y)=\mu $ has a solution. In particular, we use characterization in terms of continued fractions of $ %%
\alpha $ of all fractions $ a/b$ satisfying the inequality

$\displaystyle \left\vert \alpha -\frac{a}{b}\right\vert <\frac{2}{b^{2}}.
$

Bibliography

1
Andrej Dujella and Borka Jadrijevič.
A parametric family of quartic Thue equations.
Acta Arith., 101(2):159-170, 2002.

2
Andrej Dujella and Borka Jadrijevič.
A family of quartic Thue inequalities.
Acta Arith., 111(1):61-76, 2004.

3
Nikos Tzanakis.
Explicit solution of a class of quartic Thue equations.
Acta Arith., 64(3):271-283, 1993.

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