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In this work we introduce the numerical constant, LOPI, N LOPI is congruentto LOPI mod 18, equal to the lowest odd partition identity in conjunction withthe reduced residue system Modulo 18, a complete disjoint covering residuesystem when considered in its whole set of residues from 0 to 17. Byconvolution of specific LOPI Dirichlet linear progressions for each LOPIsuperset, the non-prime elements of the 6 reduced residue congruence classesmod 18 are generated sequentially and uniquely. Through set generation andcomplete composite number subset generation based on the pattern of sequentialfactor generation for all composite numbers in each LOPI congruence class,multiplicity becomes a possible tool to define degree of primality. We showthat the lowest sum of the digits of an integer, even or odd, is a constantunrestricted partition identity contained each natural number and is equal toone of the residues in Mod 18. We term this constant the lowest odd or evenpartition identity or LOPI or LEPI. When integers are viewed as partitions ofthe core LOPI-LEPI value, the infinite natural numbers represent the residuesof 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 arenot generated through convolution of specific Dirichlet linear progressionsinto the LOPI non-prime subset matrices, revealing the pattern of the primes tobe actually six cyclic patterns. In querying the multiplicity in these 3,4specific non prime matrices we suggest another definition of prime andnon-prime based on this multiplicity, where 0 is Prime, 1 is a secondary primeand so on. An Ox, Ox3 efficient deterministic algorithm is attached describingthe process of factor generation and primality testing.

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