Fundamental Theorem Of Arithmetic Research Articles

Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first <i>n</i> primes, <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>, sufficient conditions are given for expressing primes greater than <i>p<sub>n</sub></i>, and coprimes with prime factors greater than <i>p<sub>n</sub></i>, as algebraic functions of <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.

On the Tower Factorization of Integers

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa ; factoring each exponent a as a product of prime powers qb , and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer k ≥ 1 , we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

Algebraic Representation of Primes by Hybrid Factorization

The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.

"Al-Farisi and the Fundamental Theorem of Arithmetic, Ibn al-Hythem and Wilson Theorem (Historical Notes)"

"Al-Farisi and the Fundamental Theorem of Arithmetic, Ibn al-Hythem and Wilson Theorem (Historical Notes)"

Topological Explorations on the Fundamental Theorem of Arithmetic

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

A polynomial‐time algorithm for simple undirected graph isomorphism

SummaryThe graph isomorphism problem is to determine two finite graphs that are isomorphic which is not known with a polynomial‐time solution. This paper solves the simple undirected graph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial‐time solution to check if two simple undirected graphs are isomorphic or not. Three new representation methods of a graph as vertex/edge adjacency matrix and triple tuple are proposed. A duality of edge and vertex and a reflexivity between vertex adjacency matrix and edge adjacency matrix were first introduced to present the core idea. Beyond this, the mathematical approval is based on an equivalence between permutation and bijection. Because only addition and multiplication operations satisfy the commutative law, we propose a permutation theorem to check fast whether one of two sets of arrays is a permutation of another or not. The permutation theorem was mathematically approved by Integer Factorization Theory, Pythagorean Triples Theorem, and Fundamental Theorem of Arithmetic. For each of two n‐ary arrays, the linear and squared sums of elements were respectively calculated to produce the results.

English

English

Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of $\mathbb{N}$ on $l^2(\mathbb{N})$ is studied in association with the multiplicative structure of $\mathbb{N}$. Then the maximal ideal space of the Banach algebra generated by $\mathbb{N}$ is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The $C^\ast$-algebra generated by $\mathbb{N}$ does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by $\mathbb{N}$ is $B(l^2(\mathbb{N}))$, the set of all bounded operators on $l^2(\mathbb{N})$. Moreover, the differential operator on $l^2(\mathbb{N},\frac{1}{n(n+1)})$ defined by $\nabla~f=\mu\ast~f$ is considered, where $\mu$ is the Mobius function. It is shown that the spectrum $\sigma(\nabla)$ contains the closure of $\{\zeta(s)^{-1}:~{\rm~Re}(s)>1\}$. Interesting problems concerning $\nabla$ are discussed.

Universal cell type identifier based on number theory.

Cell type classification and handling is a key issue for understanding biological systems. The advent of high multiplexing technologies increased the complexity of the classification process and new tools are needed to support the organization of this knowledge. Ipropose a classification based on both prime numbers and the fundamental theorem of arithmetic. As a not limiting example, I show the application of this method to unambiguously define any existing cell type using the CD nomenclature established by the Human Leukocyte Differentiation Antigens Workshops. This system allows for the unique identification of any possible combination of markers hence any cell population without previous knowledge and without the need to increment the system. This method can be the future basis of any database and ontology system dealing with cell types and beyond the biological field applies to the description of any entity characterized by a list of discrete qualities. © 2018 International Society for Advancement of Cytometry.

Construction of Quasi‐Cyclic LDPC Codes Based on Fundamental Theorem of Arithmetic

Quasi‐cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5G enhanced mobile broadband (eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.

A fundamental theorem of modular arithmetic

The Fundamental Theorem of Arithmetic is a statement about the uniqueness of factorization in the ring of integers. The notation and proof easily generalize to uniqueness of factorization in principal ideal domains. Factorization, unique and otherwise, has been well-studied in integral domains. Less well-studied is factorization in rings that contain nonzero zerodivisors. In this article we give a theorem analogous to the Fundamental Theorem of Arithmetic, but for rings of the form D/(n) where D is a principal ideal domain and n is a nonzero nonunit of D.

Is This the Easiest Proof That nth Roots are Always Integers or Irrational?

SummaryMost proofs that nth roots of positive integers are always integers or irrational involve concepts such as prime numbers, relative primeness, or the fundamental theorem of arithmetic. We provide an argument which uses only the well ordering principle and the division algorithm. You can decide if it is the easiest proof possible.

Unique factorization and the fundamental theorem of arithmetic

ABSTRACTThe fundamental theorem of arithmetic is one of those topics in mathematics that somehow ‘falls through the cracks’ in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like ‘every natural number can be broken down into a product of primes’. The fact that this breakdown always results in the same primes is viewed as ‘obvious’. The purpose of this paper is to illustrate with a number of examples that the ‘Unique Factorization Property’ is a rare property and the fact that the natural numbers possess this property is ‘fundamental’ to our understanding of this number system.

Proof of the Beal Conjecture through the Fundamental Theorem of Arithmetic

This paper gives a proof of the Beal conjecture through the Fundamental Theorem of Arithmetic.

The Integer Solutions of Diophantine Equation Χ<sup>2 </sup>− 21 = 4y<sup>5</sup>

In this paper, we studied the integer solutions of the typical Diophantine equations with some important theories in quadratic fields and the fundamental theorem of arithmetic in the ring of quadratic algebraic integers. We proved all the integer solutions of the Diophantine equation Χ2 − 21 = 4y5.

A Tale of Two Monoids: A Friendly Introduction to Nonunique Factorizations

SummaryArithmetic sequences are among the most basic of structures in a discrete mathematics course. We consider here two particular arithmetic sequences: 1, 5, 9, 13, 17, … (H) and 4, 10, 16, 22, 28, …. (M)In addition to their additive definitions, these sequences are also multiplicatively closed. We show that both have multiplicative structures much different than that of the regular system of the integers. In particular, both fail the celebrated Fundamental Theorem of Arithmetic. While this is relatively easy to see, we will show that while factoring elements in the set H is fairly straightforward, factoring elements in M is much more complicated. This gives us a glimpse of how systems that fail the Fundamental Theorem of Arithmetic are studied and analyzed.

A bijection between rooted trees and fermionic Fock states: grafting and growth operators in Fock space and fermionic operators for rooted trees

We show that fermionic Fock states in the occupation number representation can be indexed uniquely by rooted trees. Our main ingredients in this construction are the Matula numbers, the fundamental theorem of arithmetic, and a relabeling of fermionic quantum states by natural numbers. As a byproduct of the correspondence mentioned above we realize the grafting and the growth operators, comprising important constructions in the context of Hopf algebras, in the fermionic Fock space. Also, we show how to construct fermionic creation and annihilation operators in the context of rooted trees. New representations of the solutions of combinatorial Dyson–Schwinger equations and of the antipode in the Connes–Kreimer Hopf algebra of rooted trees related to the occupation number picture are presented.

An All-Inclusive Proof of Beal’s Conjecture

This paper presents a complete and exhaustive proof of the Beal Conjecture. The approach to this proof uses the Fundamental Theorem of Arithmetic as the basis for the proof of the Beal Conjecture. The Fundamental Theorem of Arithmetic states that every number greater than 1 is either prime itself or is unique product of prime numbers. The prime factorization of every number greater than 1 is used throughout every section of the proof of the Beal Conjecture. Without the Fundamental Theorem of Arithmetic, this approach to proving the Beal Conjecture would not be possible.

Some integer formula encodings and related algorithms

We investigate using Sage [5] the special class of formulas made up of arbitrary but finite combinations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such formulas, we describe two essentially distinct families of canonical formula encodings for integers, respectively deduced from the decimal encoding and the fundamental theorem of arithmetic. Our main contribution is the detailed description of two algorithms which efficiently determine the canonical formula encodings associated with relatively large sets of consecutive integers.

Matula numbers, Gödel numbering and Fock space

By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a Godel type relabelling of quantum states, we construct a bijection between rooted trees and vectors in the Fock space. As a by product of the aforementioned correspondence (rooted trees $$\leftrightarrow $$ Fock space) we show that the fundamental theorem of arithmetic is related to the grafting operator, a basic construction in many Hopf algebras. Also, we introduce the Heisenberg–Weyl algebra built in the vector space of rooted trees rather than the usual Fock space. This work is a cross-fertilization of concepts from combinatorics (Matula numbers), number theory (Godel numbering) and quantum mechanics (Fock space).