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bibkey: phothila2026centered authors: Chaninat Phothila; Natthakan Thoket; Narakorn Rompurk Kanasri year: 2026 title: “Length of the Longest Arithmetic Progressions in a Certain Reduced Residue System” doi: 10.5281/zenodo.18154061 url: https://math.colgate.edu/~integers/aa6/aa6.pdf claim: “The unnumbered conjecture on printed page 14 gives the exact longest arithmetic-progression length for an even squarefree modulus with at least three prime factors when the complementary factor is less than twice the greatest prime factor.” strata_touched:

  • D5/S3/ArithUnits/CenteredReducedResidueProgressions license: citation-only triage: anchor

Centered reduced residue progressions

The paper studies arithmetic progressions contained in a fixed centered system of reduced residues. For an even modulus n, that system uses the integer representatives from -n/2+1 through n/2 that are coprime to n.

The unnumbered conjecture on printed page 14 concerns squarefree n with at least three distinct prime factors. If p is the greatest prime factor and d=n/p<2p, it predicts that the maximum progression length is floor(p-2p/d). The progression has positive step, and the maximum assertion includes both attainment and a bound for every such progression.

The formal result uses a total supremum definition for the greatest prime factor, then proves from the stated domain that its prime-factor set is nonempty. It retains the fixed centered representatives, positive natural steps, natural lengths, and the literal rational floor in the conjecture. The result is canonically frozen with statement identity ed91b7bf43b9864cca5816d00ba09b81ed5b209556c21e13b8f0c7ec5744df16, and its Scribe theorem node carries the typed Proved resolution claim for centered-reduced-residue-progressions.

Verified locator

  • Journal PDF: https://math.colgate.edu/~integers/aa6/aa6.pdf
  • DOI: 10.5281/zenodo.18154061
  • Location: printed page 14, unnumbered conjecture.

The related 2018 work of Pongsriiam addresses a different maximum problem for representative systems and is not used as a settlement of this conjecture. The bounded source and literature check is recorded in preregistration issue 8677, and the arithmetic route credits the released handoff in issue 7333. No claim of exhaustive literature coverage, worldwide priority, or independent first discovery is made.