Thue inequalities with a small number of primitive solutions

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Abstract

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. [1], [11], [6] and [2]. 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 [1], [11], [6] and [4], 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 [3], [6], [13] and [4]. 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 [1] and [6].

Original languageEnglish
Pages (from-to)199-209
Number of pages11
JournalPeriodica Mathematica Hungarica
Volume42
Issue number1-2
Publication statusPublished - Dec 1 2001

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Keywords

  • Number of solutions
  • Thue equations
  • Thue inequalities

ASJC Scopus subject areas

  • Mathematics(all)

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