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

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 2 m - 2 k n ) is either divisible by n or by a specific divisor of n . Applying this result, we construct an infinite family of certain tupl...

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Bibliographic Details
Published inThe Ramanujan journal Vol. 60; no. 4; pp. 913 - 923
Main Authors Chakraborty, Kalyan, Hoque, Azizul
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2023
Springer Nature B.V
Subjects
Combinatorics
Field Theory and Polynomials
Fields (mathematics)
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Prime numbers
11R29
11E04
11B39
Class number
Primitive divisor
Lehmer sequence
Quadruples of imaginary quadratic fields
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Summary: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 2 m - 2 k n ) 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 ( 4 d + 1 ) , Q ( 2 d + 4 ) , Q ( 2 d + 16 ) , ⋯ , Q ( 2 d + 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-022-00672-3