The primary objective of this work is to develop an accelerated version of the method of trial division, which is the simplest of the numerous attempts employed in the determination of the primality of a number. By folding the odd number line to form a matrix it is readily seen that there are infinite sets of numbers which cannot possibly be prime, that there are primes which can never be members of a set of twins and it is not possible that three (or more) consecutive odd numbers can be prime. The modified version of trial division developed here is used in conjunction with a simple formula which contains only the row and column numbers of the matrix. The sequence of operations of which this method is comprised yields either the primality of any number of interest or its factors and is eminently suitable for computerisation.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Analysis
- Array
- Discussion
- Appendix
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The main objective of this work is to improve upon the simplest method of determining primality, known as trial division. It aims to accelerate this method by rearranging the line of odd numbers into a matrix, identifying sets of numbers that cannot be prime, and utilizing a formula to simplify the process.
- Developing an accelerated version of trial division
- Identifying sets of numbers that cannot be prime
- Utilizing a matrix representation of odd numbers
- Presenting a method for determining primality suitable for computerization
- Exploring the potential of the matrix to contain all prime numbers, excluding 2
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter introduces the concept of prime numbers and Euclid's theorem on their infinitude. It explains that the work focuses on improving the trial division method, a simple approach for determining primality. The work is applicable to any odd number and is not limited to specific types of primes.
- Analysis: This chapter explores the properties of odd numbers and their arrangement in a matrix. It demonstrates that every fourth odd number is divisible by 3 and cannot be prime. The matrix representation, with its specific column structure, helps identify numbers that cannot be prime, significantly reducing the range for primality investigation.
Schlüsselwörter (Keywords)
Prime numbers, trial division, accelerated method, primality testing, matrix representation, odd numbers, prime number calculator, computerization.
- Quote paper
- William Fidler (Author), 2021, Determining of the primality of a number by the use of an accelerated version of trial division, Munich, GRIN Verlag, https://www.grin.com/document/1127210