[
  {
    "id": "ARR-2026-0P0HZ24A6Z9A1A20-v1",
    "type": "article",
    "title": "Exact small values of m(N,3) for arithmetic progressions with every common difference (Green's Problem 37)",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          22
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-0P0HZ24A6Z9A1A20",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-0P0HZ24A6Z9A1A20/versions/v1/",
    "note": "Research preprint; ARR-2026-0P0HZ24A6Z9A1A20 v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "Let m(N,k) be the least size of a finite set of integers containing, for every d=1,...,N, a k-term arithmetic progression of common difference d (Problem 37 in Green's list; k=3 unless stated). I could not find a table of its values in the literature. With CP-SAT in Z I obtain m(N,3)=3,4,5,6,7,8,8,8,9,10,11,11,12,12,12,13,13 for N=1,...,17. For N<=10 this is also supported by DRUP proofs of SAT instances whose reduction to a finite window is proved in the note; for N=11..17 the lower bounds rest on CP-SAT without a certificate. For N<=27 verified witnesses give upper bounds. The finite window [0,2N] is not harmless: for N=15 the true value is 12 but no 12-element set fits in [0,30]. A few values for k=4 are given. Everything is computational, produced by AI agents, without human review or formal verification of the pipeline."
  }
]