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Rational Diophantine sextuples with strong pair

Andrej Dujella, Matija Kazalicki, Vinko Petričević

Abstract

A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le m$, then we say the $m$-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.

Rational Diophantine sextuples with strong pair

Abstract

A set of distinct nonzero rationals such that is a perfect square for all , is called a rational Diophantine -tuple. If in addition, is a perfect square for , then we say the -tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.
Paper Structure (6 sections, 7 theorems, 31 equations)

This paper contains 6 sections, 7 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Induced elliptic curves and overview of DKMS
  3. Experiments and regularity
  4. Proof of Theorem \ref{['thm:1']}
  5. Diophantine triples with isomorphic elliptic curves
  6. Another view on families $\mathcal{F}_i$

Key Result

Theorem 1

If $(u,v)\in C(\mathbb{Q})$, then each triple $\mathcal{F}_i(u,v)$ is a rational Diophantine triple (provided that all the elements are defined, distinct and nonzero), whose first two elements form a strong Diophantine pair. Moreover, each such $\mathcal{F}_i(u,v)$ can be extended to a rational Diop

Theorems & Definitions (12)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Definition 1
  • Proposition 3
  • proof
  • Proposition 4
  • Remark 2
  • Proposition 5
  • Proposition 6
  • ...and 2 more