A procedure is developed for determining the primality of a number, N, but does not examine the number, rather a function of that number.
The function has several properties, by means of which many of the non-primes may be identified and discarded without further investigation.
Using the concept of cage numbers (defined in the text) a matrix of all of the counting numbers is produced, where, with the exception of the prime numbers 2 and 3, all of the prime numbers are embedded in one column of the matrix. Strings of consecutive numbers are extracted from the matrix, and it is seen that the only places within the whole of the range of the counting numbers where prime numbers can exist is at the positions of the second and penultimate numbers within a string
By means of the function of N, all of the odd numbers may be generated and examined for primality.
Table of Contents
Introduction
Analysis
Cage numbers and the Magic Matrix
Strings and things
The key to locating the prime numbers
Examining odd numbers for integer square roots and primality
Discussion
Objectives and Topics
The work explores a novel methodology for identifying prime numbers by analyzing their distribution within a structured matrix system rather than relying solely on traditional examination methods. The primary objective is to demonstrate that prime numbers exist at predictable locations, effectively simplifying the process of distinguishing primes from composite numbers through mathematical functions and string analysis.
- Mathematical matrix construction ("Magic Matrix")
- Definitional framework for "Cage numbers"
- Analysis of prime number distribution in primary strings
- Verification methods for primality and integer square roots
Excerpt from the Book
Analysis
In order that the reader may have sufficient data from which he/she may test the methods developed here we present sets of data taken from [2].
If we stack the odd number line in the manner shown below it is seen that we have formed a matrix having 15 columns and, assuming that the odd numbers extend to infinity, will have an infinite number of rows. The numbers shown underlined and in bold are prime numbers obtained using a prime number calculator available on Wikipaedia. Inspection of the array shows that, with the exception of columns which have a prime number as their first entry, there can never be prime numbers in columns 1, 2, 4, 7, 10, 12, and 13. Any number in the array (and the extension thereof to infinity), the sum of whose digits is divisible by 3 is not prime and can be rearranged to yield other numbers which cannot be prime; in the context of this work, which is entirely devoted to the prime numbers, any rearranged number must not end in an even number and any number, rearranged or otherwise which ends in a 5 is a member of one of columns 2, 7, or 12 and is not prime. It should be noted that the ‘spacing’ between adjacent rows is 30, whilst that between adjacent columns is 2.
Summary of Chapters
Introduction: Provides the motivation for the study, highlighting the historical significance of prime number identification and referencing key mathematical inquiries.
Analysis: Introduces a matrix-based approach for organizing odd numbers to identify patterns and constraints on where prime numbers can appear.
Cage numbers and the Magic Matrix: Defines "cage numbers" and demonstrates how grouping natural numbers results in a matrix that reveals non-random relationships between integers.
Strings and things: Defines "primary strings" of numbers and explains how prime candidates are isolated within specific positions of these strings.
The key to locating the prime numbers: Establishes rules for using the 6x table and associated numeric properties to narrow down the potential location of prime numbers.
Examining odd numbers for integer square roots and primality: Develops a concrete methodology and theorem to test for integer square roots and prime factors.
Discussion: Reviews the findings, asserting that prime numbers follow a structured distribution pattern that allows for universal application in testing for primality.
Keywords
Prime numbers, Magic Matrix, Cage numbers, Primary strings, Primality testing, Integer square roots, Mathematical analysis, Distribution of primes, Number theory, 6x table, Factors, Arithmetic, Euclidean plane, Numeric patterns, Composite numbers.
Frequently Asked Questions
What is the core focus of this publication?
The work focuses on a new perspective regarding the determination of prime numbers by identifying structured patterns in their distribution.
What are the primary thematic fields addressed?
The study covers number theory, specifically focusing on matrix structures, prime number generation, factors, and primality testing methods.
What is the primary objective of this research?
The goal is to demonstrate that prime numbers are not distributed randomly but appear at specific, identifiable locations that can be utilized for testing primality.
Which mathematical methodology is employed?
The author uses a matrix-based approach, involving "cage numbers," the construction of "primary strings," and trial division based on properties of the 6x table.
What topics are covered in the main section?
The main sections detail the construction of the "Magic Matrix," the definition of strings of numbers, and the application of these concepts to verify if a number is prime or possesses an integer square root.
Which keywords characterize this work?
Core keywords include Prime numbers, Magic Matrix, Cage numbers, Primary strings, Primality testing, and Integer square roots.
What is the significance of the "Magic Matrix"?
The "Magic Matrix" is used to confine prime numbers to specific columns, identifying relationships that simplify the distinction between prime and composite numbers.
How does the author determine if a number has an integer square root?
The author establishes a theorem proving that if N is an odd number and a specific relation of consecutive numbers does not satisfy the expression derived, then N does not possess an integer square root.
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- William Fidler (Autor:in), 2023, A new perspective on the determination of the prime numbers, München, GRIN Verlag, https://www.grin.com/document/1360641
