Periodicity and Phase of Prime Factors: Number-Theoretic Structure in the Modulo-30 Orbits and Its Potential for Post-Quantum Cryptography

Based on the eight residue classes coprime to 30 (the 8‑orbit structure), this paper reveals that every prime number greater than 5 and every prime factor of any composite number possesses a fixed period (equal to the prime factor itself) and an initial phase (determined by the residue class modulo 30). However, to obtain these phase information, one must first determine whether a number in the orbit is prime, or factor the composite number to extract its prime factors. This fact uncovers a deep relation between “phase” and “factorability” in number theory. Taking Orbit 1 (30k+1) as an example, numerical verification up to 10^8 shows that in Orbit 1 primes account for 21.6%, semiprimes for 41.65%, and other composites for 36.75%, and every prime factor obeys a fixed periodic distribution. This discovery has significant strategic value – whoever first masters the complete theory of prime periods and phases will take a leading role in the design of post‑quantum cryptography, providing a new mathematical tool for upgrading RSA‑type cryptosystems to the post‑quantum era.