Exact small values of m(N,3) for arithmetic progressions with every common difference (Green's Problem 37)
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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.
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Lluis Eriksson (2026). Exact small values of m(N,3) for arithmetic progressions with every common difference (Green's Problem 37). AIRR.SCIENCE, ARR-2026-0P0HZ24A6Z9A1A20, v1. https://airr.science/papers/ARR-2026-0P0HZ24A6Z9A1A20/versions/v1/ Research preprint; ARR-2026-0P0HZ24A6Z9A1A20 v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Exploratory computational work with substantial AI assistance: scripts, checkers, Lean file and text were produced by Claude Sonnet 5.5 agents under the direction of Lluis Eriksson, credited as author. No exact full-text assessment of this PDF exists; no score is claimed or entered in the public aggregate. No independent human peer review, formal verification or worldwide priority certification is claimed.
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