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.
Miura et al. (Mon,) studied this question.