Several upper bounds are known for the numbers of primitive solutions (x, y) of the Thue equation (1) |F(x, y)| = m and the more general Thue inequality (3) 0 < |F(x,y)| ≤ m. A usual way to derive such an upper bound is to make a distinction between "small" and "large" solutions, according as max(|x|, |y|) is smaller or larger than an appropriate explicit constant Y depending on F and m; see e.g. , ,  and . As an improvement and generalization of some earlier results we give in Section 1 an upper bound of the form cn for the number of primitive solutions (x, y) of (3) with max(|x|, |y|) ≥ Y0, where c ≤ 25 is a constant and n denotes the degree of the binary form F involved (cf. Theorem 1). It is important for applications that our lower bound Y0 for the large solutions is much smaller than those in , ,  and , and is already close to the best possible in terms of m. By using Theorem 1 we establish in Section 2 similar upper bounds for the total number of primitive solutions of (3), provided that the height or discriminant of F is sufficiently large with respect to m (cf. Theorem 2 and its corollaries). These results assert in a quantitative form that, in a certain sense, almost all inequalities of the form (3) have only few primitive solutions. Theorem 2 and its consequences are considerable improvements of the results obtained in this direction in , ,  and . The proofs of Theorems 1 and 2 are given in Section 3. In the proofs we use among other things appropriate modifications and refinements of some arguments of  and .
|Number of pages||11|
|Journal||Periodica Mathematica Hungarica|
|Publication status||Published - Dec 1 2001|
- Number of solutions
- Thue equations
- Thue inequalities
ASJC Scopus subject areas