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Computing Jacobsthal’s Function on Primorials: Exact Values and Constructive Bounds

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Alben Weeks ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

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

Provenance statement

Google Gemini assisted with C++ optimization and preparation of an earlier manuscript. OpenAI ChatGPT assisted with development, debugging, and optimization of the solver; preparation of regression tests and certificate-verification scripts; mathematical and editorial review; and drafting and revision of the exposition. These uses included code and text generation. The tools are not authors. Alben Weeks is the sole author and takes full responsibility for the mathematical claims, software, computations, references, and final text. Independent arithmetic verification in this article means checking certificates with separate integer arithmetic routines that do not use the native search engine; it does not mean independent human peer review. AI assistance is not evidence for a mathematical claim. The supplied certificates, verification routines, mathematical arguments, and explicitly scoped exhaustive-search records provide the stated support for the results

References

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Version history

  1. v1Submitted by Alben WeeksInitial depositCurrentOct 29, 2026