Deterministic Prime via Even–Odd Decomposition: Ladhe's Conjecture.

Abstract This paper proposes a conjecture on deterministic prime generation using a structured decomposition of integers into multiple even-valued components and a single odd-valued component. Specifically, we define a function P (n) = sum (Eᵢ (n) ) + O (n), where each Eᵢ (n) is even and O (n) is odd. We conjecture that there exists a class of such deterministic functions that produces prime numbers over a large domain without requiring probabilistic primality testing or rejection mechanisms. Empirical observations suggest consistent primality, motivating further theoretical investigation. ---------------------------------------------------------------------------------------------------------- This work presents a novel conjecture in number theory based on the decomposition properties of large integers into structured additive components. The paper introduces a systematic framework for representing integers as a combination of multiple terms with controlled parity and distribution characteristics. Through extensive computational examples (over 1600+ validated cases), the conjecture demonstrates consistent structural patterns across a wide numerical range. A key contribution of this work is the formulation of constraints that govern how integers can be partitioned into components with specific properties, offering potential connections to additive number theory and parity-based analysis. The results suggest deeper underlying regularities in integer decomposition, opening pathways for further theoretical investigation and potential formal proof. This submission includes detailed examples, structured datasets, and reproducible construction methods to support verification and future research.