Investigating the Subadditivity of the Prime Counting Function π(z) and Its Implications for the Second Hardy–Littlewood Conjecture

This paper investigates the subadditive properties of the prime counting function π(z) and its relationship with the Second Hardy–Littlewood Conjecture, which suggests that the prime counting function satisfies the inequality π(x + y) ≤ π(x) + π(y). We analyze this conjecture through an exploration of specific properties of prime k-tuples and their interaction with the prime numbers. The main result is an attempted proof that no integer, especially odd numbers, satisfies all three conditions simultaneously, which is required to disprove the Second Hardy–Littlewood Conjecture. This result highlights the inherent subadditivity of π(z), contributing to a deeper understanding of the conjecture. A formal proof and analysis are provided, based on insights from related literature and computational experiment, demonstrating the non-existence of integer values that fulfill all the conditions.