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Dear Colleagues,

The aim and scope of this Special Issue is to publish new results in algebraic number theory and analytic number theory, namely in ramification theory in algebraic number fields, class field theory, arithmetic functions, L-functions, modular forms, and elliptic curves. We also aim to publish results in related research directions including associative algebras, logical algebras, elementary number theory, combinatorics, difference equations, group rings, and algebraic hyperstructures.

Dr. Diana Savin
Dr. Nicusor Minculete
Dr. Vincenzo Acciaro
Guest Editors

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Keywords

ramification theory in algebraic number fields
quadratic fields
biquadratic field
cyclotomic fields
Kummer fields
Dedekind rings
p-adic fields
class field theory
elliptic curves
L-functions
modular forms
quaternion algebras
arithmetic functions
difference equations
Fibonacci, Lucas, Pell, and Horadam numbers and quaternions
logical algebras
algebraic hyperstructures
cryptography
Diophantine equations
additive number theory
partitions

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Published Papers (2 papers)

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Research

53 pages, 649 KiB

Open AccessArticle

A Group Theoretic Approach to Cyclic Cubic Fields

by Siham Aouissi and Daniel C. Mayer

Mathematics 2024, 12(1), 126; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010126 - 29 Dec 2023

Viewed by 771

Abstract

Let

{(k_{μ})}_{μ = 1}^{4}

be a quartet of cyclic cubic number fields sharing a common conductor

c = p q r

divisible by exactly three prime(power)s,

p, q, r

. For those components of the quartet [...] Read more.

Let

{(k_{μ})}_{μ = 1}^{4}

be a quartet of cyclic cubic number fields sharing a common conductor

c = p q r

divisible by exactly three prime(power)s,

p, q, r

. For those components of the quartet whose 3-class group

{Cl}_{3} (k_{μ}) ≃ {(Z / 3 Z)}^{2}

is elementary bicyclic, the automorphism group

M = Gal (F_{3}^{2} (k_{μ}) / k_{μ})

of the maximal metabelian unramified 3-extension of

k_{μ}

is determined by conditions for cubic residue symbols between

p, q, r

and for ambiguous principal ideals in subfields of the common absolute 3-genus field

k^{*}

of all

k_{μ}

. With the aid of the relation rank

d_{2} (M)

, it is decided whether

M

coincides with the Galois group

G = Gal (F_{3}^{\infty} (k_{μ}) / k_{μ})

of the maximal unramified pro-3-extension of

k_{μ}

. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications, 2nd Edition)

15 pages, 279 KiB

Open AccessArticle

Plane Partitions and a Problem of Josephus

by Mircea Merca

Mathematics 2023, 11(24), 4996; https://0-doi-org.brum.beds.ac.uk/10.3390/math11244996 - 18 Dec 2023

Viewed by 706

Abstract

The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the [...] Read more.

L (n, k)

in which you should stand in order to be the last survivor. Let

J_{n}

be the order in which the first person is executed on counting when

k = 2

. In this paper, we consider the sequence

{(J_{n})}_{n ⩾ 1}

in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving

J_{n}

. Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors’ functions and

J_{n}

are provided in this context. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications, 2nd Edition)

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