In previous work we have shown conclusively that the prime numbers can only exist at particular locations in the range of the counting numbers. A method of investigating the primality of the numbers at these locations is developed here and uses only multiplication tables. A single equation is derived which, in a sense bifurcates into two slightly different forms upon the making of one term explicit.
We can even extend the simplest definition of a prime number expressed in [5] by the addition of the phrase ‘ and can only possibly be found in the range of the counting numbers at positions on either side of any number which is divisible by 6 ’.
Table of Contents
1. Introduction
2. Analysis
3. The determination of the primality of a number
4. Discussion
5. References
Research Objectives and Themes
This work aims to present a novel, non-traditional method for investigating the primality of numbers by identifying specific patterns in their distribution. By leveraging a multiplicative formula derived from a structured matrix (the "Magic Matrix"), the author seeks to locate candidate numbers for primality and confirm their status without relying on traditional sieve methods or standard trial division.
- Identification of prime number locations relative to multiples of six.
- Development of a "pseudo prime number generating function" based on simple multiplication.
- Practical demonstration of the method through numerical examples and analytical sequences.
- Critique of existing complexity in prime-searching algorithms versus the proposed simple approach.
Excerpt from the Book
The Magic Matrix
It is conjectured that extending the matrix indefinitely will show that the prime numbers are embedded in the third row of the matrix and can only be located on either side of a number which is divisible by 6. Whilst this verifies the prime number generating function, given by the formula, p = 6 q ± 1, q = 1, 2, 3, etc it also reveals that the function is not infallible in the prediction of the prime numbers and, in the opinion of the author should more correctly be described as the pseudo prime number generating function. Nevertheless, what we have established is that we need only investigate numbers in the locations described, for all of the other numbers cannot be prime.
Summary of Chapters
Introduction: Provides a historical context to the study of prime numbers and introduces the author's alternative approach to identifying potential prime locations.
Analysis: Details the observation of prime distribution within the "Magic Matrix" and establishes the necessity of investigating numbers flanking multiples of six.
The determination of the primality of a number: Develops the mathematical application of the proposed formula and provides step-by-step numerical examples verifying the primality of specific integers.
Discussion: Evaluates the robustness of the multiplicative method and reflects on its implications for future prime number research and mathematical education.
References: Lists the cited scholarly works and sources utilized for the foundational arguments presented in the study.
Keywords
Prime numbers, Magic Matrix, Multiplication tables, Primality testing, Counting numbers, Pseudo prime generating function, Divisibility, Mathematical arithmetic, Numerical analysis, Prime locations.
Frequently Asked Questions
What is the core focus of this research paper?
The paper explores a simplified, purely multiplicative approach to identifying potential locations for prime numbers within the range of counting numbers.
Which specific areas of mathematics are primarily addressed?
The study centers on number theory, specifically focusing on the distribution patterns of prime numbers and their relationship to multiples of six.
What is the primary goal of the author's method?
The goal is to determine the primality of numbers by using a specific multiplicative formula that avoids traditional, more complex sieve methods or standard trial division.
What scientific methodology is utilized in this paper?
The author uses a heuristic approach, developing a "Magic Matrix" to isolate patterns and then applying a multiplicative derivation to test specific number candidates for primality.
What topics are covered in the main body of the work?
The main body covers the analysis of prime distribution, the derivation of a prime-generating formula, and a series of numerical examples demonstrating the efficacy of the method with test cases like q=16, q=8, and q=30.
Which keywords best describe the essence of this study?
Key terms include Prime numbers, Magic Matrix, Primality testing, Multiplicative formula, and Divisibility.
How does the author define a "pseudo prime number generating function"?
The author uses this term because while the formula identifies locations where primes might exist, it is not infallible in predicting every prime perfectly, thus requiring additional checks.
What role do multiples of six play in this study?
Multiples of six serve as the anchor points; the author demonstrates that prime numbers can only ever exist immediately to the left or right of a number divisible by six.
Is this method applicable to all prime numbers?
Yes, the author claims that the remarks and the underlying logic apply to any prime number throughout the range of counting numbers.
- Citar trabajo
- William Fidler (Autor), 2023, On the Determining of the Prime Numbers by a Simple Multiplicative Formula, Múnich, GRIN Verlag, https://www.grin.com/document/1371109
