math.NT — Number Theory
Computing Jacobsthal’s Function on Primorials: Exact Values and Constructive Bounds
Let h(x) = j(Px), where j is Jacobsthal’s function and Px is the product of the first x primes. We describe a parallel residue-covering search. Exhaustive exclusion runs, together with independently checked covering witnesses, establish h(65) = 1134, h(66) = 1160, h(67) = 1192, h(68) = 1216, h(69) = 1248, h(70) = 1274, and h(71) = 1298. Method 5B gives explicit certificates for h(100) ≥ 2054, h(150) ≥ 3558, and h(200) ≥ 5172, using partial coverings with five reserve primes. Exhaustive searches give h(100) ≤ 2798, h(150) ≤ 6898, and h(200) ≤ 11998. The search combines parity compression, descending endpoint restrictions, canonical deduplication, a sliding-window bootstrap, and dynamic prime selection related to Ziller and Morack’s Greedy Permutation Algorithm. We state the conditions needed for capacity bounds and conditional propagation, and organize extensions of complete or partial coverings in a five-tier framework. A catalog of 62 coverings at index 65 supports a verified multi-certificate envelope. The solver source, run evidence, certificates, and arithmetic verifiers are publicly available in the accompanying GitHub release v1.0-paper. Upper-bound conclusions depend on exhaustive execution of the search; the record audits do not independently repeat those searches or supply standalone unsatisfiability certificates