The Number That Refused to Add Up

We built MathOS to never claim more than it can prove. Its first verified field report: a needle hidden in five billion numbers — a fingerprint we can't yet explain — and a sequence that appears in no encyclopedia. Mathematical record (for the archive): For odd n > 3, n = p + 2^a + 2^b with p prime and a, b ≥ 0 (2⁰ = 1 allowed), the least exception is 1,117,175,145, verified exhaustively to 5 × 10⁹; exactly nine exceptions occur in (3, 5 × 10⁹], each independently confirmed, and all nine are divisible by 255 = 3 · 5 · 17. Verification extended 2026-09-04: the 255-law holds for every known term of A156695 greater than 5 — all 233 terms through 10¹² checked, zero violations — and the zero-exponent variant forms a sequence — 206 terms to 10¹² — not present in the OEIS as of 2026-09-04. Author: Sean Allen (Vanguard Creative) · Prepared by: MathOS · Date: 2026-09-03 License: CC-BY-4.0 · Venue: Zenodo preprint · Series: Field Reports from the Conjecture Pool Why we're publishing this We announced MathOS — a proof-aware engine that ingests mathematical objects and relationships, generates structure, and hunts for hidden relations. This is our first published field report, and it follows the machine's one non-negotiable rule: every claim wears its evidence status, or it doesn't get made.FORMALLY_VERIFIED. COMPUTATIONALLY_TESTED. EMPIRICALLY_SUPPORTED. CONJECTURED. CONTRADICTED. No default stamps, no "trust me." A relation earns its label or wears the honest one. Below is a true story from that engine's first months: a computation that found something real, a pattern it refuses to call a proof, and a sequence that lives in no published encyclopedia.