Even–Odd Decomposition of Prime Numbers: Ladhe's Conjecture

Abstract This paper proposes a conjectural framework for analyzing prime numbers based on a structured decomposition of integers into multiple even-valued components and a single odd-valued component. We define a function of the form:P(n) = Σ Eᵢ(n) + O(n),where each Eᵢ(n) is even and O(n) is odd.By construction, P(n) is always odd. For those values of n where P(n) is prime, this expression provides a structured even–odd decomposition of a prime number.We investigate whether there exists a class of such deterministic constructions for which prime values arise systematically. Unlike traditional approaches that verify primality after candidate generation, this framework explores whether primes exhibit an inherent structural representation through such decompositions.Empirical observations across a large dataset indicate consistent emergence of such decomposition patterns among prime numbers under specific configurations, motivating further theoretical investigation.