Partitioning the Primes and Efficient Non-trivial Factor Generation using the Least Odd Partition Identity in Conjunction with Dirichlet Linear Progressions and the Reduced Residue System Modulo 18

TL;DR

This paper introduces the LOPI constant and its relation to prime and composite number generation using residue systems, Dirichlet progressions, and a new multiplicity-based prime definition, supported by an efficient factorization algorithm.

Contribution

It presents the LOPI constant and a novel residue-based framework for understanding prime distribution and factor generation, including a new prime classification based on multiplicity.

Findings

Primes form six cyclic patterns within the residue system.

The LOPI system enables sequential and unique generation of non-prime numbers.

A deterministic algorithm for factorization and primality testing is provided.

Abstract

In this work we introduce the numerical constant, LOPI, N LOPI is congruent to LOPI mod 18, equal to the lowest odd partition identity in conjunction with the reduced residue system Modulo 18, a complete disjoint covering residue system when considered in its whole set of residues from 0 to 17. By convolution of specific LOPI Dirichlet linear progressions for each LOPI superset, the non-prime elements of the 6 reduced residue congruence classes mod 18 are generated sequentially and uniquely. Through set generation and complete composite number subset generation based on the pattern of sequential factor generation for all composite numbers in each LOPI congruence class, multiplicity becomes a possible tool to define degree of primality. We show that the lowest sum of the digits of an integer, even or odd, is a constant unrestricted partition identity contained each natural number and is equal to one of the residues in Mod 18. We term this constant the lowest odd or even partition identity or LOPI or LEPI. When integers are viewed as partitions of the core LOPI-LEPI value, the infinite natural numbers represent the residues of Mod 18 as extended each time toward infinity by sequential cycles of 18. LOPI mod 18 is a factor generation system that demonstrates why the primes are not generated through convolution of specific Dirichlet linear progressions into the LOPI non-prime subset matrices, revealing the pattern of the primes to be actually six cyclic patterns. In querying the multiplicity in these 3,4 specific non prime matrices we suggest another definition of prime and non-prime based on this multiplicity, where 0 is Prime, 1 is a secondary prime and so on. An Ox, Ox3 efficient deterministic algorithm is attached describing the process of factor generation and primality testing.