Algebraic Number Research Articles

The development of P-adic numbers theoretically and their use in number theory

Stemming from the need to formalize the divisibility of rational numbers, p-adics provide a unique mathematical perspective rooted in prime number theory. This article provides a comprehensive discussion of p-adic numbers, including their research background, definition, properties, theory, and extensions. This study first elucidates the historical background and significance of p-adic numbers, emphasizing their key role in number theory and its applications. At the heart of this research is a deep dive into the precise definition of p-radical numbers, revealing their unique, often counterintuitive, distance measure. We look at their fundamental characteristics, demonstrate their illogical characteristics, and discuss how they affect transcendental number theory, Diophantine equations, and algebraic number theory. Furthermore, this article explores the theoretical foundations of p-radical numbers and their extensions, emphasizing their integral role in advanced mathematical structures such as p-radical analysis and p-radical geometry. By covering these aspects, this study aims to highlight the lasting impact of p-adic on modern mathematics, reshape our understanding of divisibility, and advance mathematical inquiry into new and uncharted territory.

Some arithmetical properties of convergents to algebraic numbers

Some arithmetical properties of convergents to algebraic numbers

Primes in denominators of algebraic numbers

Denote the set of algebraic numbers as [Formula: see text] and the set of algebraic integers as [Formula: see text]. For [Formula: see text], consider its irreducible polynomial in [Formula: see text], [Formula: see text]. Denote [Formula: see text]. Drungilas, Dubickas and Jankauskas show in a recent paper that [Formula: see text]. Given a number field [Formula: see text] and [Formula: see text], we show that there is a subset [Formula: see text], for which [Formula: see text]. We prove that [Formula: see text] is a principal ideal domain if and only if the primes in [Formula: see text] generate the class group of [Formula: see text]. We show that given [Formula: see text], we can find a finite set [Formula: see text], such that for every number field [Formula: see text], we have [Formula: see text]. We study how this set [Formula: see text] relates to the ring [Formula: see text] and the ideal [Formula: see text] of [Formula: see text]. We also show that [Formula: see text] satisfy [Formula: see text] if and only if [Formula: see text] for all number fields [Formula: see text].

Euclidean volumes of hyperbolic knots

The hyperbolic structure on a 3 3 –dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.

On Transcendental Regular Continued Fractions

ABSTRACT We establish explicit constructions of real transcendental numbers that are not U-numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.

On Repdigits Which are Sums or Differences of Two k-Pell Numbers

ABSTRACT Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,…, 0, 1 and each term afterwards is given by the linear recurrence p n ( k ) = 2 P n − 1 ( k ) + P n − 2 ( k ) + ⋯ + P n − k ( k ) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262].

Some geometric properties of the Padovan vectors in Euclidean 3-space

Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with initial conditions and general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. Up to now, various studies have been conducted on Padovan numbers and Padovan polynomial sequences. In this study, Padovan vectors are defined for the first time by using the Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm, and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in the Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

The interconnection between local and global m-th powers and Grunwald-Wang theorem

The relation between local and global solution of an equation can be discussed with the method of class field theory and algebraic number theory. In this piece of writing, the author will introduce the behavior of both local and global m-th power in some specific number field. Of course, the result in this paper can be extended into the function field, but it will not be involved in this paper. This paper will prove that if k(_(2^t ))/k is cyclic then the Local m-th powers everywhere is equivalent to the global m-th power. In the Non-cyclic case this decomposition becomes P(m,S)=k^mk^m. This paper will also prove some useful propositions in topological group theory, which will be used in the proof of Grunwald-Wang theorem. Grunwald-Wang theorem states that we can find a cyclic extension with given local behaviors. To describe the extension, this paper combines character theory with a topological group, one can depict the cyclic extension. This theorem can be used in the further exploration of central simple algebra.

The special value 𝑢=1 of Artin-Ihara 𝐿-functions

We study the special value u = 1 u=1 of Artin-Ihara L L -functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.

Geometrical aspects of the algebraic number related to quasicrystals

Geometrical aspects of the algebraic number related to quasicrystals

On Algebraic Number Theory Domains Dedekind

This article is based on a series talks I gave on the topic mentioned in the title in a Counsellor Seminar at proms’. The exposition till Section is based on and most of the remainder is based on and some notes on Algebraic Number Theory available.

ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS

AbstractLet $K={\mathbb {Q}}(\theta )$ be an algebraic number field with $\theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${\mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if $\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc.100 (2019), 239–244] conjectured that if $n\ge 3$ , $1\le m\le n-1$ , $\gcd (n,mB)=1$ and A is a prime number, then the polynomial $x^n+A (Bx+1)^m\in {\mathbb {Z}}[x]$ is monogenic if and only if $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. We prove that this conjecture is true.

ℤ-gradings on the Grassmann algebra over infinite fields: Graded identities and central polynomials

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of E equipped with its [Formula: see text] and 3-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.

A kit for linear forms in three logarithms

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds for linear forms in logarithms. We give worked examples to demonstrate both the use of our technique and the improvements it provides. Publicly shared code is also available.

On the abc$abc$ conjecture in algebraic number fields

AbstractIn this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

Solutions of the Diophantine Equations B r = J s + J t and C r = J s + J

Let B r r ≥ 0 , J r r ≥ 0 , and C r r ≥ 0 be the balancing, Jacobsthal, and Lucas balancing numbers, respectively. In this paper, the diophantine equations B r = J s + J t and C r = J s + J t are completely solved. The solutions rely basically on Matveev’s theorem on linear forms in logarithms of algebraic numbers and a procedure of reducing the upper bound due to Dujella and Pethö.

ON JOINT DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N + M] with with real algebraic numbers a1,...,ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.

Number theoretical locally recoverable codes

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.

Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F(x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F(x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study Fk,N(x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by Fk,N(x). Radchenko and Zagier gave a beautiful relation between the integral ∫01log⁡(1+tx)1+tdt and F(x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between Fk,N(x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of Fk,N(x) and some generalized Lambert series are also obtained along with other supplementary results.

On the Finiteness of the Set of Generalized Jacobians with Nontrivial Torsion Points over Algebraic Number Fields

On the Finiteness of the Set of Generalized Jacobians with Nontrivial Torsion Points over Algebraic Number Fields