NOTE ON THE <i>p</i>-DIVISIBILITY OF CLASS NUMBERS OF AN INFINITE FAMILY OF IMAGINARY QUADRATIC FIELDS

Krishnamoorthy, Srilakshmi; Pasupulati, Sunil Kumar

doi:10.1017/s001708952100015x

Cited by 6 publications

(4 citation statements)

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“…2 ) with d ∈ Z whose class numbers are all divisible by any odd integer n ≥ 3. This extends the results of [4,12,15] in both directions; from pairs to quadruples of fields and from primes to odd integers. It also gives a proof of a weaker version of Conjecture 1.1 for any prime p ≥ 3 (in fact for any odd integer n ≥ 3).…”

Section: Introductionsupporting

confidence: 81%

“…Theorem D can be viewed as a weaker variant of a generalization of Conjecture 1.1. For m = 1, it provides a generalization of the main result of [15] though [23] appeared before [15]. In other words, Theorem D gives a complete proof of the following generalization of Conjecture 1.1 for m = 1 and a proof of a weaker version of the same for m ≥ 2.…”

Section: Discussionmentioning

confidence: 83%

“…Very recently, Krishnamoorthy and Pasupulati [15] cleverly used [16,Theorem 1] and an extended version of [9,Theorem 3.2] to settled Conjecture 1.1 for m = 1.…”

Section: Introductionmentioning

confidence: 99%

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On a conjecture of Iizuka

Hoque

2021

Preprint

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For a given odd positive integer n and an odd prime p, we construct an infinite family of quadruples of imaginary quadratic fieldsthat the class number of each of them is divisible by n. Subsequently, we show that there is an infinite family of quintuples of imaginary) with d ∈ Z whose class numbers are all divisible by n. Our results provide a complete proof of Iizuka's conjecture (in fact a generalization of it) for the case m = 1. Our results also affirmatively answer a weaker version of (a generalization of) Iizuka's conjecture for m ≥ 4.

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Section: Introductionsupporting

confidence: 81%

Section: Discussionmentioning

confidence: 83%

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On a conjecture of Iizuka

Hoque

2021

Preprint

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“…Iizuka [15] himself settled the conjecture for imaginary quadratic fields for ℓ = 3 and k = 1. Recently, Conjecture 1.2 has been settled for k = 1 and for all primes ℓ in [18]. Some general cases have also been tackled in [5], [6], [14] and [26].…”

Section: Introductionmentioning

confidence: 99%

On the $p$-rationality of consecutive quadratic fields

Chattopadhyay¹,

Laxmi²,

Saikia³

2022

Preprint

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In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic p-rational number fields of degree 2 t for any odd prime number p and any integer t ≥ 1. Using the criteria provided by him to check p-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic p-rational fields have been shown to exist in recent years. In this article, for any integer k ≥ 1, we build upon the existing work and prove the existence of infinitely many prime numbers p for which the imaginary quadratic fields Q( −(p − 1)), . . . , Q( −(p − k)) and Q( −p(p − 1)), . . . , Q( −p(p − k)) are all p-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of p-rationality for two consecutive real quadratic fields by proving the existence of infinitely many p-rational fields of the form Q( p 2 + 1) and Q( p 2 + 2). The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.

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