Algebraic Number Theory

The author had published a paper on the solutions for the twin primes conjecture in an international mathematics journal in 2003. This paper approaches the twin primes problem through the analysis of the intrinsic nature of the prime numbers.

This is the ultimate proof that the Prime Numbers are not Random Numbers as famous Mathematicians believe and claim publicly through presentations you can find on U Tube
Any pupil around the world and a wide public will understand the first 2 pages of the paper below.  Feel free to ask questions by emailing me constantine.adraktas@mit-partners.eu

This paper contains a new proof of Euler’s theorem, that the only non-trivial integral solution, (α, β), of α2 = β3 +1 is (±3, 2). This proof employs only the properties of the ring, Z, of integers without recourse to elliptic curves and is independent of the methods of algebraic number fields. The advantage of our proof, over Euler’s isolated and other known proofs of this result, is that it charts a common path to a novel approach to the solution of Catalan’s conjecture and indeed of any Diophantine equation. Mathematics Subject Classification: 11R04

This paper expounds the role of the non-trivial zeros of the Riemann zeta function ζ and supplements the author’s earlier papers on the Riemann hypothesis. There is a lot of mystery surrounding the non-trivial zeros. MSC: 11-XX (Number Theory)

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

The paper considers tilings of Euclidean plane where the set of prototiles contains three shapes: ♢(72° rhombus), ⧫(36° rhombus), and ⬠(the regular pentagon); all having the same side length. Some tilings by these 3 shapes are locally derivable from well-known Penrose tilings, and two different presentations of a Penrose tiling are described. Certain matching rules are necessary for Penroseanness in each of the two cases. They are not sufficient in one presentation and are sufficient in the other. Rest of the paper is preoccupied mainly with finding similar and related systems of prototiles and matching rules (admitting Penrosean tilings, but not exclusively), as well as checking the Penrosean property for an arbitrary tiling. Some derivability results about certain class of tilings are presented. Several yet unsolved problems are formulated.
There is also a distinct “algebraic” section approaching the five-fold symmetric geometry from integral algebraic numbers and the ring theory.  It results in a formalism which is usually explained as a four-dimensional lift of a tiling.

Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange. This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra. This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography. In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given ; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.

Euclid’s proof of the infinitude of the primes has generally been regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which will always bring in larger and larger primes, an infinite number of them. However, the proof is also subtle and has been misinterpreted by some with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. The author has been working on the twin primes conjecture for a long period and had published a paper on the conjecture in an international mathematics journal in 2003. This paper presents some remarks/reasons which support the validity of the twin primes conjecture, including a reasoning which is somewhat similar to Euclid’s proof of the infinity of the primes. (This paper is published in an international mathematics journal.)

The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.

This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.

Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.

This is an introdution to discrete valuation rings (DVR), a particular kind of ring used algebraic number theory and algebraic geometry. The essay is divided in three sections. The …rst section covers the elementary de…nitions, examples and properties of DVRs. The second section is focused in the proof of the Main Theorem of DVRs which characterizes them as Noetherian rings with particular properties. Finally last section introduces an important example of DVR: the p-adic integers.

This is my masters dissertation, completed under the supervision of Dr. Tobias Berger.

I give an introduction to elliptic curves with a view to proving that the group of rational points is finitely generated. Throughout I use the "cannonball problem" as motivation.

Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.It is usually denoted by the Greek letter  (phi).

In this paper we present some peculiarities related to representation of numbers when the language used in the speech is the Brazilian Sign Language (Libras). We will address some characteristics of the formation of numbers in Libras and the syntactic structure of the language. The existence of two systems of cardinal numbers in Libras and signs of regional variations. Differences between the form of numerical representation of Arabica System for the spoken or written languages in oral and Libras. Influences the form of numerical representation in understanding the concepts transmitted. Concluding with the need for research on how to express themselves in Libras themes of mathematics education.

Let E be an elliptic curve of conductor pq^2, where p and q are prime numbers, and let K be a quadratic extension of Q. If K is imaginary and p and q are split in K, there are Heegner points on the modular curve X_0(pq^2) defined over ring class
fields attached to orders in K, which can be mapped to points on E. If q is inert, there are no Heegner points on the modular curve, but points can be obtained from the Cartan non-split curve X^ε_ns(q, p). If K is real the panorama is quite different. If
p is inert and q is split, Stark-Heegner points have been defined on E, whose field of definition is conjecturally the narrow ring class field attached to an order in K. This work combines these two ideas, defining Stark-Heegner points when p and q are inert
in the real quadratic field K, using Cartan non-split curves, which are conjecturally defined over narrow ring class fields attached to orders in K.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. MSC: 11-XX (Number Theory)

The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.

This is a quite informal view of global class field theory, viewed from the platform of ideals.

See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.

In this paper we analyse the construction of p−adic numbers, which we expand to the new combinatorial p − q−adic numbers. Thus, introducing combinatorics into algebraic number theory. After we validate the made progress, the topology and the algebra, we construct new functions, which are embedded infinite (combinatorial − and or periodic tower of) fields. Before we come to categories and functors, we also introduce briefly p−q−adic m−nomial combinatorial functions, respectively − series and give an example of a distribution function for it. Further, we show how a p−q−adic distribution function might be constructed.

In this article we describe some elements from the chapter of Mathematics " Number Theory " , as they appear in the Codex Vindobonensis phil. Gr. 65, a Byzantine Ms kept in the National Library of Austria in Vienna. This codex contains a comprehensive program of teaching Mathematics which was addressed to an audience consisting of students probably of all the grades of what is today's primary and secondary education, but also state functionaries, merchants, craftsmen of various specialities such as silversmiths and goldsmiths, builders, etc. The mathematical branch of Number Theory and the numerical position system is integrated in codex 65 in the respective chapters of the four operations and their checks. Introduction At the beginning of this article we briefly describe Codex Vindobonensis phil. Gr. 65 fols 11r ―126r, which was written circa AD 1436 by an anonymous author, and was intended for teaching purposes. We record the syllabus units so that the reader will form an initial idea about the content of the Ms, and then we analyze in detail the problems related to the Number Theory, but also more generally to the numerical system of position in Byzantium.

This Master's thesis, 'Norms of Ideals in Direct Sums of Number Fields and Applications to the Circulants Problem of Olga Taussky-Todd', presents original research.  As the title suggests, to tackle a problem which was originally posed by Olga Taussky-Todd, who asked what values can be taken by the determinant of a matrix with integer entries where each row is a cyclic shift of the previous one.  Hitherto algebraic results for specific cases have been proven by M. Newman using matrix manipulation.  To address the general case, the approach here uses a well-known relationship between determinants of matrix transformations and 'absolute' norms of fractional ideals in a direct sum of number fields, using asymptotic methods.  This gives an indication of what proportion of integers are values.  Some additional new proofs are given for results already presented by Newman.

The research was undertaken at the University of Glasgow, with the thesis originally submitted in 1992.  The author has belatedly made the work available here, with some corrections, in the hope that it may be of some value for other researchers and especially to honour the memory of his supervisor, the late Prof. Robert (“Bob”) W. K. Odoni.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes

Since being officially selected as the new Advanced Encryption Standard (AES), Rijndael has continued to receive great attention and has had its security continuously evaluated by the cryptographic community. Rijndael is a cipher with a simple, elegant and highly algebraic structure. Its selection as the AES has led to a growing interest in the study of algebraic properties of block ciphers, and in particular algebraic techniques that can be used in their cryptanalysis. In these notes we will examine some algebraic aspects of the AES and consider a number of algebraic techniques that could be used in the analysis of the cipher. In particular, we will focus on the large, though surprisingly simple, systems of multivariate quadratic equations derived from the encryption operation, and consider some approaches that could be used when attempting to solve these systems. These notes refer to an invited talk given at the Fourth Conference on the Advanced Encryption Standard (AES4) in May 2004, and are largely based on[4].

The Prime Numbers do not spring out randomly but follow a neatly structured system The Prime Numbers, bar number 2 and 3, reside in the [ 6α + 5 ] and [ 6α + 7 ] Matrices
The Prime and Composite Numbers in [ 6α + 5 ] and [ 6α + 7 ] give the impression of having a random pattern when indeed they have a neatly structured pattern Primes and Composites in the Primes Residencies

The concept of different ideal is important in algebraic number theory be- cause it encodes the ramification data in extension of algebraic number fields. In this article, we wish to characterize the different ideal geometrically using the notion of Kahler differential and hence giving a way for it to fit into higher dimensional algebraic geometry.