Primes as Sums of Consecutive Composites: Classification and a Modulo-12 Constraint

Abstract

This work introduces a constructive framework representing primes as sums of consecutive composites. Two representation types are defined: Type1 (Pure Representation): A prime p is expressed as the sum of consecutive composite numbers. Type2 (Corrected Representation): A prime p is expressed as the sum of consecutive composites plus one composite within the window. The Liu Conjecture proposes that every prime p > 13 admits at least one of these two representations. Through exhaustive enumeration of all consecutive composite intervals up to 10^8, we obtained a complete classification of all 5,761,449 primes greater than 13. The results confirm Liu's Conjecture within this range. A striking arithmetic constraint is discovered: all 948 exclusively Type2 primes (those without any Type1 representation) strictly satisfy p ≡ 1 or 11 (mod 12). Their distribution is: 491 in class 1, 457 in class 11, and zero in classes 5 and 7. This repository contains the experimental datasets, classification results, and analysis code supporting the findings. Keywords: prime numbers, composite sequence, consecutive composites, Liu's Conjecture, modulo-12 constraint, classification, computational number theory