Number Theory

   

Prime Set Representation

Authors: Steven Shawcross

The integer 2 satisfies the divisibility definition of a prime number: it is only divisible by itself and 1. The integer 1 also satisfies this definition, and yet, mathematicians generally do not consider 1 a prime. Rather 1 merits a class of its own, belonging neither to the prime nor composite class. In divisibility theory, 2 does occupy a special subclass within the class of prime numbers: it is the only even prime. This paper introduces a theory of numbers called the Prime Set Representation Theory. This theory utilizes the odd primes and does not rely on the primeness of 2. In Prime Set Representation Theory, the odd primes are building blocks of the theory; all integers, including 2, have representations in terms of them. The import of the theory is not to dislodge the integer 2 from its solitary, even-prime status. The theory's efficacy is a better understanding of the distribution of primes, twin primes, and primes of the form x^2 + 1. A natural extension of the theory yields valid and strikingly direct approximation formulas for these prime classifications. The same theory furnishes a new and improved approximation to the number of Goldbach pairs associated with general even number 2n (the improvement is relative to Sylvester's formula for Goldbach pairs, but the formula performs well vis-à-vis the Hardy-Littlewood formulas in the ranges tested).

Comments: 9 Pages. A version of this paper is copyrighted by Steven Shawcross, 2003.

Download: PDF

Submission history

[v1] 2017-10-17 09:44:37

Unique-IP document downloads: 12 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus