Simultaneous Convergent Continued Fraction Algorithm for Real and p-adic Fields with Applications to Quadratic Fields

2023/09/18 by Shin-ichi Yasutomi, Yasutomi, Shin-ichi · 1 citation
Mathematics · #11D88 #11J61 #11J70 #11Y65 #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2309.09447

openalex publication_date 2023/09/18 · openalex created_date 2023/09/20 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number and K be a field with embeddings into ℝ and ℚ_p. We propose an algorithm that generates continued fraction expansions converging in ℚ_p and is expected to simultaneously converge in both ℝ and ℚ_p. This algorithm produces finite continued fraction expansions for rational numbers. In the case of p=2 and if K is a quadratic field, the continued fraction expansions generated by this algorithm converge in ℝ, and they are eventually periodic or finite. For an element α in K, let p_n/q_n denote the n-th convergent. There exist constants u₁ and u₂ in \mathbb R_>0 with u₁ + u₂ = 2, and constants C₁ and C₂ in \mathbb R_>0 such that |α- p_n/q_n| < C₁/|q_n|^u₁ and |α- p_n/q_n|₂ < C₂/|q_n|^u₂. Here, |⋅|₂ represents the 2-adic distance. For prime numbers p > 2, we present numerical experiences.

Cited by

• The arithmetic of continued fractions in the field of p-adic numbers

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