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

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

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

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

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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