On the divisibility of the class number of the imaginary quadratic field ℚ ( $$\sqrt {a^2 - 4k^n } $$ )

1989 ◽  
Vol 5 (1) ◽  
pp. 80-86
Author(s):  
Le Maohua
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field

A certain eta-quotient and the class number of an imaginary quadratic field

2014 ◽  
Vol 10 (06) ◽  
pp. 1485-1499
Author(s):  
Takeshi Ogasawara
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Theta Series ◽  
Imaginary Quadratic Field

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals THE DIVISIBILITY OF THE CLASS NUMBER OF THE IMAGINARY QUADRATIC FIELD

2011 ◽  
Vol 54 (1) ◽  
pp. 149-154 ◽  
Author(s):  
ZHU MINHUI ◽  
WANG TINGTING
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Positive Integers

AbstractLet hK denote the class number of the imaginary quadratic field $K=\mathbf{Q}(\sqrt{2^{2m}-k^n})$, where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

scholarly journals Limiting equations for the class number of the ideals of an imaginary quadratic field

1965 ◽  
Vol 5 (2) ◽  
pp. 303-305
Author(s):  
O. Saparnijazov
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: О. Сапарниязов. Асимптотические равенства для числа классов идеалов мнимого квадратического поля O. Saparnijazovas. Menamojo kvadratinio skaičių kūno idealų klasių skaičiaus asimptotinės išraiškos

An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Veronika Trnková
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Relative Class ◽  
Relative Class Number ◽  
Stickelberger Ideal ◽  
Imaginary Field ◽  
Divisibility Properties

AbstractWe consider certain extension of the Stickelberger ideal of the compositum of a bicyclic field and a quadratic imaginary field, obtained by adding new annihilators to the Stickelberger ideal. We compute the index of this extension, from which we get some divisibility properties for the relative class number of the compositum.

scholarly journals A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Garcia Villeda
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Dirichlet Character ◽  
Imaginary Quadratic Field ◽  
Kronecker Symbol ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Computable Formula

<p style='text-indent:20px;'>Using elementary methods, we count the quadratic residues of a prime number of the form <inline-formula><tex-math id="M2">\begin{document}$ p = 4n-1 $\end{document}</tex-math></inline-formula> in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> of the imaginary quadratic field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document}</tex-math></inline-formula> Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.</p>

scholarly journals ON THE DIOPHANTINE EQUATION x2 + d2l + 1 = yn

2012 ◽  
Vol 54 (2) ◽  
pp. 415-428 ◽  
Author(s):  
ATTILA BÉRCZES ◽  
ISTVÁN PINK
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Squarefree Integer

AbstractLet d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.

scholarly journals Averages and moments associated to class numbers of imaginary quadratic fields

2017 ◽  
Vol 153 (11) ◽  
pp. 2287-2309 ◽  
Author(s):  
D. R. Heath-Brown ◽  
L. B. Pierce
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Upper Bounds ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Higher Moments ◽  
Imaginary Quadratic Fields

For any odd prime $\ell$, let $h_{\ell }(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_{\ell }(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_{\ell }(-d)$ for all primes $\ell \geqslant 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geqslant 3$.

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