TITLE:
A Structural-Decomposition and Asymptotic Framework toward a Formal Proof of Lemoine’s Conjecture
AUTHORS:
Duncan Ndegwa, Loyford Njagi, Stephen Luketero, Benard Nzimbi, Kikwai Benjamin
KEYWORDS:
Lemoine’s Conjecture, Prime Partitions, Semiprimes, Asymptotic Density, Additive Number Theory, Recursive Partitioning
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
26,
2026
ABSTRACT: Lemoine’s conjecture asserts that every odd integer
O>5
can be written in the form
O=p+s
, where
p
is an odd prime and
s
is an even semiprime. Despite extensive computational verification, a general proof remains open. This paper develops a structural framework for the conjecture based on additive decompositions of odd integers. We introduce a combinatorial reformulation in which Lemoine representations arise as constrained elements within a broader class of odd integer partitions. A recursive partitioning algorithm is then proposed to systematically generate and test candidate prime-semiprime decompositions, yielding a well-defined counting function
f(
O
)
. Heuristic density considerations, based on classical estimates for primes and semiprimes, suggest that
f(
O
)
is typically large, although no unconditional asymptotic lower bound is established. The framework further situates Lemoine’s conjecture within a wider class of additive problems involving primes and almost-primes, and provides a conditional reduction into computational verification and distributional analytic components.