A method is devised in which the prime number counting function, pi(k) is viewed as having the properties of a staircase. The terminology appropriate to a staircase is used to describe the parts of the distribution amongst the integers.
Strings of limited extent of consecutive numbers which may contain prime numbers are generated from an iterative equation which contains Gauss' prime number counting function. In honour of Gauss we call these strings of numbers, Gauss strings. , and they are used in the construction of the staircase.
Further, we show that the prime number counting function may be represented in number space by an infinite set of connected trapezia, the individual areas of which are numerically equal to the gap between the primes that are situated on two of the borders of any given trapezium.
Table of Contents
1. Introduction
2. Analysis.
3. The Gauss string
4. The determination of the primality of a number
5. The simple geometry associated with a tread
6. Discussion
7. References
Objectives & Core Themes
This work proposes a novel method to represent the prime number counting function as a geometric object by treating it as a staircase structure. The primary objective is to develop an iterative approach ("Gauss strings") to construct the prime number distribution and to demonstrate a geometrically based analytical framework for determining the primality of numbers.
- Geometric representation of the prime number counting function.
- Development of the iterative "Gauss string" formula for prime distribution.
- Application of an accelerated trial division method for primality testing.
- Geometric interpretation of prime gaps as areas of connected trapezia.
Excerpt from the Book
The simple geometry associated with a tread
Fig6 shows a tread and riser with the leading edge of the tread connected by a construction line of length, L to the leading edge of the next higher tread.
Let θ be the angle between the line L and the line p1------ p2, and since we are considering a tread, then the ‘p’ are prime numbers. The line of length, x, is located at the next odd number after, p1.
Now, tan θ = x/2 = 1/(p2 - p1), hence, p2 - p1 = 2/x; but, 2/x must be an even integer. From this it follows that we must set, x = 1/n, where, n = 1, 2, 3, 4, -------∞.
Hence, p2 - p1 = 2n, which is the gap between the primes and, tan θ = 1/2n. The triangle may now be shown with one side equal to unity whilst the other two sides are simple functions of n.
Summary of Chapters
Introduction: Provides a historical context on Riemann's formula for predicting prime numbers and its significance in relation to number theory.
Analysis.: Establishes the conceptual model of the prime number counting function as a staircase where risers possess unit height.
The Gauss string: Introduces the iterative formula used to generate sequences of integers where primes can be located.
The determination of the primality of a number: Presents an accelerated version of trial division to determine the primality of remaining candidates within the generated sequences.
The simple geometry associated with a tread: Details the geometric mapping of prime gaps onto triangles and trapezia to visualize the distribution.
Discussion: Offers a critical reflection on fitting a discontinuous prime counting function with continuous functions.
References: Lists the academic sources and works cited throughout the text.
Keywords
prime number counting function, Gauss string, primality testing, staircase model, number theory, prime distribution, iterative formula, trial division, geometric objects, prime gaps, rectangular hyperbola, complex analysis, integers, Riemann, mathematical synthesis.
Frequently Asked Questions
What is the primary focus of this work?
The work focuses on synthesizing the prime number counting function by modeling it as a "staircase" and developing efficient, iterative geometric methods to track and identify prime numbers.
What are the core themes explored in the text?
Central themes include the conceptualization of prime distributions as staircase structures, the generation of "Gauss strings" to locate primes, and the use of geometric identities to represent gaps between primes.
What is the main research goal?
The goal is to demonstrate that the distribution of prime numbers can be constructed and represented through geometric objects and iterative algebraic formulas without relying on standard approximations.
Which scientific methodology is primarily employed?
The methodology relies on an iterative mathematical formula to create strings of consecutive integers and combines this with an accelerated method of trial division to verify primality.
What is covered in the main body of the text?
The main body covers the conceptual staircase model, the formal definition of Gauss strings, a practical demonstration of primality testing, and the geometric mapping of prime gaps.
Which keywords best characterize this publication?
Key terms include: Prime number counting function, Gauss string, primality testing, geometric representation, and iterative distribution method.
What is a "Gauss string"?
A Gauss string is a range of consecutive integers generated by an iterative formula based on an initial even number, designed to provide a limited set in which prime numbers can be systematically located.
How is the gap between primes interpreted geometrically?
The gap between primes is interpreted as the area of connected trapezia or as a function of the angle within a triangular construction in number space.
What is the significance of the "staircase" terminology?
It provides a physical analogy for the discontinuous nature of the prime number counting function, where treads represent sequences of integers and risers represent the jumps in the function.
What criticism does the author raise in the discussion?
The author criticizes the traditional effort to approximate a discontinuous function (like the prime number counting function) with continuous functions, arguing that the latter introduces "meaningless" regions where the function is undefined.
- Citar trabajo
- William Fidler (Autor), 2023, On the Synthesis of the Distribution amongst the Integers of the Prime Number Counting Function, pi(k), viewed as a Geometric Object, Múnich, GRIN Verlag, https://www.grin.com/document/1336494
