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Quadratic residues and domino tilings

Yuhi Kamio, Junnosuke Koizumi, Toshihiko Nakazawa

Abstract

The formula for the number of domino tilings due to Kasteleyn and Temperley-Fisher is strikingly similar to Eisenstein's formula for the Legendre symbol. We study the connection between these two concepts and prove a formula which expresses the Jacobi symbol in terms of domino tilings.

Quadratic residues and domino tilings

Abstract

The formula for the number of domino tilings due to Kasteleyn and Temperley-Fisher is strikingly similar to Eisenstein's formula for the Legendre symbol. We study the connection between these two concepts and prove a formula which expresses the Jacobi symbol in terms of domino tilings.
Paper Structure (9 sections, 18 theorems, 72 equations)

This paper contains 9 sections, 18 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Domino tilings
  4. Norms
  5. Algebraic Proof
  6. Kasteleyn operator
  7. Realization in an algebra
  8. Computation of the norm
  9. Combinatorial proof

Key Result

Theorem 1.1

Let $m,n$ be positive integers and assume that $n$ is odd. Then we have Here, the right hand side is the Jacobi symbol.

Theorems & Definitions (44)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 34 more