A cyclic structure of prime numbers modulo 9

This note presents an observation on the distribution of prime numbers modulo 9.

Every prime number p ≥ 5 is congruent to 1, 2, 4, 5, 7 or 8 modulo 9. These six classes form two distinct cycles:

    Cycle A: 1 → 7 → 4 → 1
    Cycle B: 2 → 8 → 5 → 2

Two arithmetic properties are demonstrated:

1. Within the same class modulo 9, the gap between two consecutive primes is a multiple of 18: p_{n+1} = p_n + 18k.

2. To move from one class to the next within the same cycle (1→7, 7→4, 4→1 or 2→8, 8→5, 5→2), the gap is a multiple of 6: q = p + 6k.

A connection with famous prime families is also shown:
    - Mersenne primes (2^n - 1) belong to Cycle A.
    - Fermat primes (2^(2^n) + 1) belong to Cycle B.

An experimental validation on 18,000 candidates (3,000 per class) around 30,000 confirms a uniform distribution of primes among the six classes (6,051 primes found, maximum deviation of 0.17% from uniformity).