Statement
For all prime gaps g = p_{n+1} − p_n with p_n > 3, we have g ≡ 0, 2, or 4 (mod 6). Equivalently, no prime gap after the initial gap of 1 is congruent to 1, 3, or 5 modulo 6.
Proof
Every prime p > 3 satisfies p ≡ 1 (mod 6) or p ≡ 5 (mod 6) the only residues modulo 6 that are coprime to 6 and strictly greater than 3. The four possible cases for (p mod 6, q mod 6), where q is the next prime after p, are therefore:
- (1, 1) → q − p ≡ 0 (mod 6)
- (1, 5) → q − p ≡ 4 (mod 6)
- (5, 1) → q − p ≡ 2 (mod 6)
- (5, 5) → q − p ≡ 0 (mod 6)
All four cases yield gaps ≡ 0, 2, or 4 (mod 6). ∎
Empirical verification
The theorem has been verified computationally for all prime gaps up to one million. Across 348,511 consecutive prime gaps, zero violations of the Mod-6 Gap Constraint were observed. The only gap not congruent to 0, 2, or 4 modulo 6 is the initial gap of 1 between 2 and 3 the boundary case excluded by the theorem's hypothesis p_n > 3.
Gap 2 ≡ 2 (mod 6) - "twin-type" transitions
Gap 4 ≡ 4 (mod 6) - "twin-complement" transitions
Gap 6 ≡ 0 (mod 6) - same-residue-class break
Gaps 8, 10, 12, … - all ≡ 0, 2, or 4 (mod 6)
Related results
Theorem VII is closely related to Theorem VIII (Twin Prime Hex-Spacing): the gap between consecutive twin prime pairs (p, p+2) and (q, q+2) with p, q > 3 is always divisible by 6. The proof proceeds similarly: p > 3 in a twin prime pair forces p ≡ 5 (mod 6) (since p ≡ 1 would make p + 2 ≡ 3 composite), so q ≡ 5 as well, and q − p ≡ 0 (mod 6).
Within the Tesfa Grid geometry, this constraint expresses itself as a structural separation between the residue classes that primes can occupy and those that they cannot see Theorem IV (Row-6 Structural Prime Exclusion).
Cite this result
Dereje, T. (2026). The Tesfa Grid: A Deterministic Harmonic Sieve and Its Structural Interactions with Prime Numbers, Volume I, Theorem VII (Mod-6 Gap Constraint). Section 8.1.