On an indivisibility version of Iizuka's conjecture

2024/11/13 by R. Muneeswaran, R, Muneeswaran, Srilakshmi Krishnamoorthy +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.08772

openalex publication_date 2024/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Iizuka's conjecture predicts that, given m ∈ ℕ and a prime p, there exists infinitely many integers n such that the class numbers of all of the following quadratic number fields, ℚ(√(n)), ℚ(√(n+1)), …, ℚ(√(n+m)), are divisible by p. In this article, given k and m, we study the proportion of n such that the class numbers of none of the successive fields ℚ(√(n)), ℚ(√(n+1)), …, ℚ(√(n+m)), are divisible by \( 3^k \). Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by 3.

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