In this paper, a mathematical structure is formed which consists of a matrix of all of the odd numbers. It then follows that the matrix contains all of the prime numbers with the exception of 2. In conjunction with a prime number counting function denoted by pi(x) we can determine the number of primes in any row of the matrix and their location using the prime number search method developed, which uses the prime number counting function data compiled from various sources by A V Kulsha.
The columns of the matrix are of infinite extent and in some there are either one or no prime numbers. In a column where there is a single prime number, this without exception appears in the first row. The matrix shows that there are prime numbers which can never be twinned and that it is not possible, with the exception of 3,5,7 that three consecutive odd numbers can be prime.
Further it is demonstrated, in some of the rows, even although the distribution of the primes is random, that it is possible to predict the next prime, and in some instances the prime following it, thus predicting a set of twins. Moreover in many cases the gap between consecutive primes can be predicted.
The work, in particular the Discussion contains numerous examples . Any odd number can, in principle be located precisely within the matrix. In the Appendix it is shown how, using only the prime number counting function data, to determine the primality (or otherwise) of any odd number without examining forensically the number itself.
It is noted that, in contradistinction to formulae which predict particular prime numbers, such as those of Mersenne and Sophie Germain, the analysis presented here makes no distinction between any of the 'types' of primes.
Inhaltsverzeichnis (Table of Contents)
- Abstract
- Introduction
- Analysis
- Formulae for the prediction of the number of primes in any row of the matrix
- A local prime number density
- Discussion
- Appendix
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work investigates the distribution of prime numbers within a matrix formed by the odd numbers. It seeks to demonstrate a method for predicting the number of primes in any given row of the matrix, as well as their location. The work aims to reveal patterns and insights into the seemingly random distribution of prime numbers.
- Decomposition of the odd number line into a matrix
- Prime number counting function specific to each row of the matrix
- Predicting prime numbers and gaps between them
- Identifying twin primes and limitations on their occurrence
- Analyzing the distribution of prime numbers within the matrix
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: Introduces the concept of prime numbers and their significance in mathematics. Discusses the challenge of understanding their distribution amongst odd numbers. Briefly outlines the approach taken in this work.
- Analysis: Presents a matrix representation of the odd numbers, with an infinite number of rows and fifteen columns. Explains how prime numbers are identified in the matrix and describes a method for determining the number of primes in any given row.
- Formulae for the prediction of the number of primes in any row of the matrix: Explores a formulaic approach to predict the number of primes in each row of the matrix. This section likely delves into the development and application of the prime number counting function.
- A local prime number density: Examines the distribution of primes within a row of the matrix. This section may analyze the density of prime numbers in different regions of the matrix, potentially revealing patterns or variations in their distribution.
- Discussion: Presents a discussion of the findings and their implications, including examples of prime number prediction and analysis of prime twin occurrences within the matrix.
Schlüsselwörter (Keywords)
Prime numbers, odd numbers, matrix representation, prime number counting function, prime number distribution, prime number density, twin primes, prime number prediction.
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- William Fidler (Autor:in), 2021, On the disposition of the prime numbers, München, GRIN Verlag, https://www.grin.com/document/1139873
