Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lucas sequences, Pell's equations, and automorphisms of K3 surfaces

Kwangwoo Lee

TL;DR

This work ties together the arithmetic of Lucas and generalized Fibonacci sequences with the geometry of K3 surfaces via automorphisms of rank-2 Picard lattices. By establishing correspondences among automorphisms, the sequences a_n and b_n, and Pell equation solutions, it provides a unified framework to describe when a number is a generalized Fibonacci term and to characterize the traces of automorphisms in terms of Pell-type data. The authors solve several Pell system problems by translating them into intersection problems of Lucas sequences, yielding explicit descriptions in terms of Lucas sequences and their discriminants. The results illuminate a deep bridge between Diophantine equations and geometric automorphisms, with potential implications for arithmetic geometry and lattice theory.

Abstract

We have the correspondences between Lucas sequences, Pell's equations, and the automorphisms of K3 surfaces with Picard number 2. Using these correspondences, we determine the intersections of some Lucas sequences.

Lucas sequences, Pell's equations, and automorphisms of K3 surfaces

TL;DR

This work ties together the arithmetic of Lucas and generalized Fibonacci sequences with the geometry of K3 surfaces via automorphisms of rank-2 Picard lattices. By establishing correspondences among automorphisms, the sequences a_n and b_n, and Pell equation solutions, it provides a unified framework to describe when a number is a generalized Fibonacci term and to characterize the traces of automorphisms in terms of Pell-type data. The authors solve several Pell system problems by translating them into intersection problems of Lucas sequences, yielding explicit descriptions in terms of Lucas sequences and their discriminants. The results illuminate a deep bridge between Diophantine equations and geometric automorphisms, with potential implications for arithmetic geometry and lattice theory.

Abstract

We have the correspondences between Lucas sequences, Pell's equations, and the automorphisms of K3 surfaces with Picard number 2. Using these correspondences, we determine the intersections of some Lucas sequences.
Paper Structure (12 sections, 13 theorems, 40 equations)

This paper contains 12 sections, 13 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lucas sequences
  4. Lattices
  5. K3 surfaces
  6. Pell's equations
  7. Generalized Fibonacci sequences
  8. Proof of Theorem \ref{['correspondence of three items']}
  9. Case $a_n$.
  10. Case $b_n$.
  11. Intersection of Lucas sequences
  12. Intersection of two $V$-sequences

Key Result

Theorem 1.1

There are the correspondences between the following sets.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 16 more