Lehmer sequence approach to the divisibility of class numbers of imaginary quadratic fields

2022/10/02 by Kalyan Chakraborty, Chakraborty, Kalyan, Azizul Hoque +1 · 1 citation
Mathematics · #11B39 #11E04 #11R29 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2210.00561

openalex publication_date 2022/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k≥ 3 and n≥ 3 be odd integers, and let m≥ 0 be any integer. For a prime number ℓ, we prove that the class number of the imaginary quadratic field ℚ(√ℓ^2m-2kⁿ) is either divisible by n or by a specific divisor of n. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form (ℚ(√(d)), ℚ(√(d+1)), ℚ(√(4d+1)), ℚ(√(2d+4)), ℚ(√(2d+16)), ⋯, ℚ(√(2d+4^t)) ) with d∈ ℤ and 1≤ 4^t≤ 2|d| whose class numbers are all divisible by n. Our proofs use some deep results about primitive divisors of Lehmer sequences.

Cited by

• On the simultaneous 3-divisibility of class numbers of quadruples of real quadratic fields

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