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Construction of diagonal quintic threefolds with infinitely many rational points

Maciej Ulas

Abstract

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over $\Q$ each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples $B=(B_{0}, B_{1}, B_{2}, B_{3})$ of co-prime integers such that for a suitable chosen integer $b$ (depending on $B$), the equation $B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b$ has infinitely many positive integer solutions.

Construction of diagonal quintic threefolds with infinitely many rational points

Abstract

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples of co-prime integers such that for a suitable chosen integer (depending on ), the equation has infinitely many positive integer solutions.
Paper Structure (4 sections, 5 theorems, 40 equations)

This paper contains 4 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Warm up
  3. The construction of a quadratic parametrization
  4. An application

Key Result

Theorem 3.1

Let $t\in\mathbb{Z}\setminus\{-1, 0, 1\}$ and put $A(t)=(A_{0}(t), \ldots, A_{4}(t))$. The variety $\mathcal{V}_{A(t)}$ contains infinitely many non-trivial integer points.

Theorems & Definitions (14)

  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • ...and 4 more