Azizul Hoque

Harish-Chandra Research Institute | HRI · Faculty of Mathematics

J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find...

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

Let c be a square-free positive integer and p a prime satisfying p∤c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\not \mid c$$\end{document}. Let h(-c)\documentclas...

Let \(a\ge 1\) and \(n>1\) be odd integers. For a given prime p, we prove under certain conditions that the class groups of imaginary quadratic fields \({\mathbb {Q}}(\sqrt{a^2-4p^n})\) have a subgroup isomorphic to \({\mathbb {Z}}/n{\mathbb {Z}}\). We also show that this family of fields has infinitely many members with the property that their cla...

For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ with $d\in \mathbb{Z}$ such that the class number of each of them is divisible by $n$. Subsequently, we show...

Let S be a certain affine algebraic surface over \({\mathbb Q}\) such that it admits a regular map to \({\mathbb A}^2/{\mathbb Q}\). We show that any non-trivial torsion element in the Chow group \({ {CH}}^1(S)\) can be pulled back to ideal classes of quadratic fields whose order can be made as large as possible. This gives an affirmative answer to...

Let $d$ be a square-free integer and $\mathbb{Z}[\sqrt{d}]$ a quadratic ring of integers. For a given $n\in\mathbb{Z}[\sqrt{d}]$, a set of $m$ non-zero distinct elements in $\mathbb{Z}[\sqrt{d}]$ is called a Diophantine $D(n)$-$m$-tuple (or simply $D(n)$-$m$-tuple) in $\mathbb{Z}[\sqrt{d}]$ if product of any two of them plus $n$ is a square in $\ma...

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for prime $n$, under the condition $n\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field...

We exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers. We also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields. At the end, we provide some numerical examples supporting our results.

We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^my^n$, where $a,b,c, m\geq 0, n \geq 3$ and $x, y\geq 1$ are unknown integers with $\gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and f...

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x, y, m, n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.

We prove that for every integer n, there exist infinitely many D(n)-triples which are also D(t)-triples for t∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in {\ma...

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ 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...

Let $d\equiv 2\pmod 4$ be a square-free integer such that $x^2 - dy^2 =- 1$ and $x^2 - dy^2 = 6$ are solvable in integers. We prove the existence of infinitely many quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ when $n \in \{(4m + 1) + 4k\sqrt{d}, (4m + 1) + (4k + 2)\sqrt{d}, (4m + 3) + 4k\sqrt{d}, (4m + 3) + (4k + 2)\sqrt{d}, (4m +...

Generalized Mersenne numbers are defined as $M_{p,n} = p^n- p + 1$, where $p$ is any prime and $n$ is any positive integer. Here, we prove that for each pair $(c, p)$ with $c\geq 1$ an integer, there is at most one $M_{p, n}$ of the form $cx^2$ with a few exceptions.

We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions o...

We completely solve the Diophantine equation x 2 + 2 k 11 ℓ 19 m = y n in integers x, y ≥ 1; k, ℓ, m ≥ 0 and n ≥ 3 with gcd(x, y) = 1, and use this to recover some earlier results in this direction.

Groups of order $4$ are isomorphic to either $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$. We give certain sufficient conditions permitting to specify the structure of class groups of order $4$ in the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$ as $n$ varies over positive integers. Further, we c...

We investigate the class number one problem for a parametric family of real quadratic fields of the form $\mathbb{Q}(\sqrt{ m^2 + 4r})$ for certain positive integers $m$ and $r$.

Let $l, p$ be odd primes, $q = p^r , r \in \mathbb{ Z}^+ , q \equiv 1 \pmod {2l^2}$ and $\mathbb{F}_q$ a field with $q$ elements. The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss but still effective formulae are yet to be known. In this pap...

It is well-known that for p=1, 2, 3, 7, 11, 19, 43, 67, 163 the class number of $\mathbb{Q}(\sqrt{-p})$ is one. We use this fact to determine all the solutions of $x^2+p^m=4y^n$ in non-negative integers $x, y, m$ and $n$ with an exception only when p=2. This will appear in 'Publicationes Mathematicae Debrecen'

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a conjecture by H. Wada \cite{WA70} on the structure of ideal class groups.

We consider the Diophantine equation $x^2+17^k41^\ell 59^m =2^\delta y^n$ in unknown integer $x\geq 1, y>1, k, \ell, m, \delta\geq 0$ and $n\geq 3$ with $\gcd(x,y)=1$, and we find all its solutions. We use the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences in combination with elementary number th...

We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be 3. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be 2 and 3.

International audience In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.

We solve the Diophantine equation x 2 + 2 k 11 19 m = y n in integers x, y ≥ 1; k, , m ≥ 0 and n ≥ 3 with gcd(x, y) = 1 and use this to recover some earlier results in this direction.

This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4–7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields a...

We construct some families of quadratic fields whose class numbers are divisible by 3. The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers are divisible by 3. At the end we compute class number of these fields for some small values and verify our resul...

We find all positive integer solutions in x, y and n of x 2 + 19 2k+1 = 4y n for any non-negative integer k.

Class numbers of quadratic fields have been the object of attention for many years and there exist a large number of interesting results. This is a survey aimed at reviewing results concerning the divisibility of class numbers of both real and imaginary quadratic fields. More precisely, to review the question ``do there exist infinitely many real (...

We investigate the solvability of the Diophantine equation x2-my2=±p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^2-my^2=\pm p$$\end{document} in integers for certa...

We produce an infinite family of imaginary quadratic fields whose ideal class groups have $3$-rank at least $2$.

Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. In [28] (Proc Int Conf–Number Theory 1, 107–116, 2004), Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta transformation formula and Gerver (Amer J Math 92, 33–55, 1970) [12] was the firs...

In this paper we introduce the concept of $\xi$-torsion module, $\xi$-torsion-free module and $\xi$--torsionable module. We investigate many properties of these modules. We characterize $\xi$-torsion modules and $\xi$-torsion-free modules using short exact sequences and module homomorphisms.

In this paper, the notion of quasi-pseudo injectivity relative to a class of submodules, namely, quasi-pseudo principally injective has been studied. This notion is closed under direct summands. Several properties and characterizations have been given. In particular, we characterize Noetherian Rings and Dedekind Domains by quasi-pseudo principally...

For any square-free positive integer m, let H(m) be the class-number of the field , where ζm is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}2 − 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}2 −2, where g is a positive integer. We also sho...

In this paper we provide criteria for the insolvability of the Diophantine equation \(x^2+D=y^n\). This result is then used to determine the class number of the quadratic field \({\mathbb {Q}}( {\sqrt{-D} })\). We also determine some criteria for the divisibility of the class number of the quadratic field \({\mathbb {Q}}( {\sqrt{-D} })\) and this r...

In this paper we introduce a secure and efficient public key cryptosystem using generalized Mersenne primes based on two hard problems: the cubic root extraction modulo a composite integer and the discrete logarithm problem (DLP). These two problems are combined during the key generation, encryption and decryption phases. To break the scheme, an at...

In this paper our attempt is to investigate the class number problem of imaginary quadratic fields. We establish that the class number of imaginary quadratic field , for certain positive integer q, is a multiple of 3. We also show that there are infinitely many imaginary quadratic fields whose class numbers are multiples of 3.

In this paper we define equivalent quadratic fields and prove that generalized Mersenne primes generate a family of infinitely many equivalent quadratic fields with equivalent index \(2\) and whose class numbers are divisible by 3. We also prove that the class-number of the cyclotomic field \(\mathbb {Q}\big ( \zeta _m \big )\), where \(m\in \mathb...

The prime numbers of the form $M_(p,q)=p^q-p+1$, where $p$ and $q$ are positive integers, we call Generalized Mersenne primes (GMPs). In this paper we study GMPs and establish some important results based on these primes in connection with some arithmetic functions. We also discuss solvability of quadratic congruences modulo $2q+1$ under certain co...

In this paper, we obtain some new results for perfect numbers and generalized perfect numbers connected with the relationship among arithmetic functions σ , φ and ψ. These arithmetic functions and their compositions play vital role in this work.