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Some congruences of calibers of real quadratic fields

Naoto Fujisawa

Abstract

In this paper, the congruence equations for caliber and m-caliber in various discriminants are proven. Additionally, We also obtained the lengths of the periods of several continued fractions as corollaries.

Some congruences of calibers of real quadratic fields

Abstract

In this paper, the congruence equations for caliber and m-caliber in various discriminants are proven. Additionally, We also obtained the lengths of the periods of several continued fractions as corollaries.
Paper Structure (4 sections, 32 theorems, 187 equations)

This paper contains 4 sections, 32 theorems, 187 equations.

Table of Contents

  1. § 1. Introduction
  2. § 2. Preliminaries
  3. § 3. Proofs of Theorems
  4. § 4. Conjecture

Key Result

Theorem 1

8p Let $p$ be a prime number such that $p \equiv 1$ (mod 4) and let $x_p$ and $y_p$ be integers satisfying $p=x_p^2+y_p^2$ and $0<x_p<y_p$. Then we have

Theorems & Definitions (45)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
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  • Theorem
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  • ...and 35 more