Additive Prime Identities Involving Three Primes: Clarification on Lemoine's Variant and a new specified Decomposition

This version clarifies the historical context of the first identity explored in the paper. The identity , where  is an odd number and  are primes, is structurally equivalent to Lemoine’s (or Levy’s) conjecture, which states that every odd number greater than 5 can be written as the sum of a prime and twice a prime. In this version, the identity is acknowledged as a variant rather than a novel formulation.The paper’s core contribution remains the second identity , where all terms are prime. This structure has not been found in the mathematical literature and appears to offer a new perspective on prime decomposition. The paper includes empirical results and Python code confirming the identity over a large prime range.