Sparacino Conjecture 2026 on Primes After the Product of Consecutive Prime Gap and 1 p 1

Let p₁ and p₂ be two consecutive primes, d = p₂ − p₁, and M = d · p₁. Define N = ⌈ln²M⌉. The Sparacino Conjecture states that the open interval (M, M+N] always contains at least one prime. Computational evidence. The conjecture has been verified with zero counterexamples across two independent GPU campaigns: Run 1 — exhaustive sieve up to p₁ ≤ 10¹³: all 139,864,010 consecutive prime pairs verified using a deterministic Miller–Rabin test (7 witnesses, CUDA, RTX GPU). Zero failures. Run 2 — window-based 128-bit CUDA verification at scales 10¹³ through 10³⁰: 74,860,909 additional pairs sampled randomly near each scale, using a 15-witness deterministic Miller–Rabin kernel. Zero failures. In total, 214,724,919 pairs were tested with no counterexample, covering seventeen orders of magnitude from p₁ = 2 to M ~ 10³⁰. Structural properties. Three lemmas are established analytically. (1) For p₁ > 2, d is always even, so M is always even and the search begins on an odd number — immediately halving the candidate set. (2) Since p₁ | M and N < p₁ for all p₁ ≥ 97 (verified and proved), none of the values M+k with k ∈ [1, N] is divisible by p₁ — eliminating an entire residue class from the sieve. (3) For every prime q dividing d, M ≡ 0 (mod q), yielding a multiplicative density correction C(d) = ∏q|d q/(q−1) ≥ 2. Together these give M = d·p₁ a structurally privileged position compared to a generic even number of the same magnitude. Statistical regularity. The ratio k̄ / ln M remains in the narrow band 1.02–1.04 across all tested scales, exhibiting empirical stability over 17 orders of magnitude. The maximum ratio k/N observed is 0.414 — the sieve window is never close to saturation. Towards a proof. A rigorous proof outline is sketched combining two components. For M ≥ 10⁷ (equivalently ln M ≥ 16), the result of Baker, Harman and Pintz (2001) guarantees a prime in (x, x + x^0.525] for all sufficiently large x; since ln²M ≪ M^0.525 for all M ≥ 10⁴, the Sparacino interval is contained in the BHP interval, yielding an unconditional proof for this range. For M < 10⁷, the 214 million pairs described above cover the range computationally with zero failures. The gap between the computational verification and the BHP asymptotic threshold is bridged by the 128-bit GPU campaign, which reaches M ~ 10³⁰ — well beyond any reasonable estimate for the BHP effective constant. Relationship to existing conjectures. Firoozbakht's conjecture implies the Sparacino conjecture logically (as a special case), but the converse does not hold: Sparacino may be provable even if Firoozbakht fails, because the points M = d·p₁ carry additional arithmetic structure that generic primes do not. The ratio (gap after the prime below M) / ln²(that prime) never exceeds 0.60 at Sparacino points, compared to a maximum of 0.97 for primes in general. Code and data. All verification scripts are released under CC-BY-SA. The original exhaustive sieve uses NumPy + CuPy with a CUDA C++ Miller–Rabin kernel. The extended 128-bit campaign uses a custom CUDA kernel with __uint128_t arithmetic and 15-witness determinism. Aggregated statistics (histograms of k, dependence on d mod q, mod 3 residues of p₁) are archived in compressed .npz format.