R. Muneeswaran
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View article: On an indivisibility version of Iizuka's conjecture
On an indivisibility version of Iizuka's conjecture Open
Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{…
View article: A collage of results on the divisibility and indivisibility of class numbers of quadratic fields
A collage of results on the divisibility and indivisibility of class numbers of quadratic fields Open
The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…
View article: Congruence classes for modular forms over small sets
Congruence classes for modular forms over small sets Open
J.P. Serre showed that for any integer $m,~a(n)\equiv 0 \pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cu…
View article: A study on `$t$-divisibility of class numbers of certain family of imaginary quadratic fields.
A study on `$t$-divisibility of class numbers of certain family of imaginary quadratic fields. Open
We prove that for a given odd number $m\geq3$, for all but finitely many primes `p', class number of $\mathbb{Q}(\sqrt{1-2m^p})$ is divisible by `p' and this collection of fields is infinite for a fixed `m'. We also prove that the class nu…
View article: The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$
The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$ Open
Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid h_{…
View article: Investors’ Behaviour on Investment Avenues
Investors’ Behaviour on Investment Avenues Open
Investors are confronted with a set of investment avenues, to spend their savings, based on the risk and returns availability. The behavior of investors would differ with reference to time, personality and specific needs. Therefore, the st…