The work developed here brings a significant measure of order to the search for prime numbers.
It is shown conclusively that no Mersenne number can have an integer square root, this being, until now, one of the unsolved problems in Number theory. It is shown that there are a new set of numbers given by F = (M + 2)/3 where M is a Mersenne number. it is found that when M is prime then F is prime. however, it is also found that prime numbers may be generated when M is composite and the index of the Mersenne number is prime, but this is not invariably the case.
An accelerated method of trial division is derived and from which we construct one row matrices which we have called, templates. These are found to be of great utility in determining factors of numbers, and their 'extent' is equal to the number of digits in a number of interest. The template may be easily extended by a simple process explained in the text. Extension of a template by one unit increases that range of numbers which may be examined by an order of magnitude.
Table of Contents
1. Introduction
2. Analysis
3. The Magic Matrix
4. Primary Strings and the Rule of 6
5. The Method of Fractions
6. Templates
7. Mersenne numbers
8. Discussion
Research Objectives and Topics
This work aims to bring a new level of order to the search for prime numbers by utilizing mathematical structures like the "Magic Matrix," the "Rule of 6," and the "Method of Fractions." The research seeks to simplify the identification of potential prime numbers, provide a novel method for accelerated trial division, and analyze the properties of Mersenne numbers within these new framework contexts.
- Mathematical disposition of prime numbers within the range of counting numbers.
- Development of the "Magic Matrix" and the "Rule of 6" for prime location identification.
- Implementation of the "Method of Fractions" for accelerated trial division and factor determination.
- Structural investigation of Mersenne numbers and their relation to specific characteristic signatures.
- Exploration of "Fidler numbers" derived from a specific formula applied to Mersenne primes.
Excerpt from the Book
The Method of Fractions
Rather than proceed to develop this concept expressed in arcane mathematical format we adopt an approach, characteristic of much of previous work by the author, and explain the process by example.
Consider the decadal sequence 10^9, 10^8, 10^7, 10^6, 10^5, 10^4, 10^3, 10^2, 10^1, 1. Let us divide each of the terms by 6 starting at the RH end of the sequence. The numerators of the remainders of this procedure will then be, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4. We call the dividing number, in this case, 6, the base number. We now choose a number, at random and show how to determine its location in a primary string.
Consider the number 1253964723. Starting from the left we now write all of the above numerators in the following manner: 4 4 4 4 4 4 4 4 4 1 and immediately beneath it the random number, 1 2 5 3 9 6 4 7 2 3.
We now form the undernoted sums. 1x4 = 4, 2x4 = 8, 5x4 = 20, 3x4 = 12, 9x4 = 36, 6x4 = 24, 4x4 = 16, 7x4 = 28, 2x4 = 8, 1x3 = 3. We now sum the results on the RHS of these simple multiplications and divide by 6, i.e. 139/6 = 23 + 1/6.
Summary of Chapters
Introduction: Provides a historical context to prime number research and outlines the transition of prime number interest from purely philosophical to practical applications in modern encryption.
Analysis: Presents data sets demonstrating the confinement of prime numbers to specific positions within primary strings and their relation to numbers divisible by six.
The Magic Matrix: Explores the conjecture that prime numbers are embedded in the third row of the matrix and argues that prime-generating functions should be considered pseudo-prime generators.
Primary Strings and the Rule of 6: Defines the "Rule of 6" as a procedure for locating numbers within primary strings using characteristic fractional signatures of 1/6 and 5/6.
The Method of Fractions: Introduces an accelerated version of trial division that utilizes decimal-based templates to identify factors of very large numbers.
Templates: Defines the implementation of matrices for fitting large numbers to templates to quickly verify divisibility by a specific base number.
Mersenne numbers: Investigate Mersenne numbers using the established concepts, demonstrating their specific locations in primary strings and their square-free nature.
Discussion: Summarizes the novel findings of the work, including the settling of the square-root conjecture for Mersenne numbers and the introduction of "Fidler numbers."
Keywords
Prime Numbers, Magic Matrix, Mersenne Numbers, Rule of 6, Method of Fractions, Primary Strings, Characteristic Signatures, Trial Division, Fidler Numbers, Number Theory, Encryption, Factors, Templates, Composite Numbers, Mathematical Conjecture.
Frequently Asked Questions
What is the core focus of this work?
The work focuses on bringing a structured approach to identifying prime numbers and evaluating properties of Mersenne numbers using new mathematical definitions.
What are the central thematic fields involved?
The core themes include number theory, the development of primality test algorithms, and the structural categorization of counting numbers.
What is the primary goal of this research?
The goal is to simplify the process of locating prime numbers and determining the factors of large numbers through systematic matrices and fractional remainder methods.
Which scientific methodology does the author employ?
The author employs a constructive, empirical approach, developing original frameworks like the "Magic Matrix" and "Method of Fractions" to verify properties of large integers.
What is covered in the main body of the text?
The main body details the construction of primary strings, the mechanics of the "Rule of 6," the development of reusable templates for trial division, and the analysis of Mersenne number square roots.
Which keywords best characterize this research?
Key terms include Prime Numbers, Mersenne Numbers, Rule of 6, Magic Matrix, and Method of Fractions.
Can any Mersenne number have an integer square root according to the author?
No, the author proves that no Mersenne number can have an integer square root, characterizing this as a settled problem within this framework.
What are the so-called "Fidler numbers"?
Fidler numbers are a sequence of numbers (F = (M+2)/3) generated from Mersenne primes, which the author observes are often prime themselves.
How does the "Rule of 6" assist in locating numbers?
It identifies the location of a number within a primary string by examining the remainder of a division by 6, providing a "characteristic signature" for that number.
Why is the "Method of Fractions" considered important?
It provides an accelerated, powerful form of trial division that avoids the complexities of long division, making it highly efficient for computer-based implementation.
- Quote paper
- William Fidler (Author), 2023, Mersenne Numbers, Strings, the Rule of 6, and the Method of Fractions, Munich, GRIN Verlag, https://www.grin.com/document/1389654
