PAX Conjecture: A Prime Addition–Multiplication Expression of Integers — Part I Intuitive Structure, Definition, and the Reason Why 10 Is the Only Exception

This note introduces the PAX Conjecture (Prime Addition × Multiplication eXpression), a new framework for representing integers using a combination of prime multiplication and prime addition. The conjecture states that every integer n ≥ 6, except for n = 10, can be expressed in the form n = p × q + r where p, q, r are primes. The products p × q act as “milestones” on the integer line, while the prime r adjusts these milestones to cover the surrounding integers. Part I presents the motivation, intuitive structure, formal definition, the structural reason for the unique exception at 10, and an introductory connection with the prime number theorem.