This paper highlights a previously unnoticed cyclic structure of prime numbers modulo 9. The six admissible residue classes 1, 2, 4, 5, 7, 8 partition into two cycles of length three: Cycle A: 1 → 7 → 4 → 1Cycle B: 2 → 8 → 5 → 2. We show that this organization follows from the Galois theory of the cyclotomic extension ℚ (ζ₉) /ℚ, and we make explicit two group actions (multiplicative and additive) that generate the spacing laws p+18k (same class) and p+6k (cycle). An interpretation using the Cayley graph of ℤ/9ℤ is proposed. These results, experimentally validated, have potential applications in cryptography (RSA key generation, elliptic curves, pairings). Terms of use: This paper is protected by copyright. Any industrial or commercial use is prohibited without the prior written consent of the author. Academic citation is permitted provided the source and DOI are acknowledged.
MONIA DAOUDI (Sat,) studied this question.