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Let \(a, Q\in \textbf{Q}\) be given and consider the set \(\mathcal {G}(a, Q)=\{aQ^{i}:\;i\in \textbf{N}\}\) of terms of geometric progression with 0th term equal to a and the quotient Q. Let \(f\in \textbf{Q}(x, y)\) and \(\mathcal {V}_{f}\) be the set of finite values of f. It is known that for any quadratic form f, there are infinitely many pairs a, Q such that \(\mathcal {G}(a, Q)\subset \mathcal {V}_{f}\) . In this note we prove that the same statement is true for reducible cubic forms. The main tool in the proof is the density of rational points on the quartic surface \(z^3=(x^2-u)(y^2-u)\) , where \(u\in \textbf{Q}\setminus \{0\}\) .

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Geometric Progressions in the Sets of Values of Reducible Cubic Forms

  • 会议论文

  • 发表日期：

    2025年

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  • Bartosz Głowacki,
  • Maciej Ulas

摘要

Let \(a, Q\in \textbf{Q}\) be given and consider the set \(\mathcal {G}(a, Q)=\{aQ^{i}:\;i\in \textbf{N}\}\) of terms of geometric progression with 0th term equal to a and the quotient Q. Let \(f\in \textbf{Q}(x, y)\) and \(\mathcal {V}_{f}\) be the set of finite values of f. It is known that for any quadratic form f, there are infinitely many pairs a, Q such that \(\mathcal {G}(a, Q)\subset \mathcal {V}_{f}\) . In this note we prove that the same statement is true for reducible cubic forms. The main tool in the proof is the density of rational points on the quartic surface \(z^3=(x^2-u)(y^2-u)\) , where \(u\in \textbf{Q}\setminus \{0\}\) .