{
  "schema_version": 1,
  "problem_number": "EP-278",
  "title": "An Exact Fixed-Parameter Algorithm for Extremal Unions of Residue Classes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let A = {n₁ < ⋯ < nᵣ} be a finite set of positive integers. Choose one residue class aᵢ mod nᵢ for each modulus and maximize the natural density of their union. This is the unsettled maximum-density half of Erdős Problem 278. We give an exact uniform characterization whose state space depends on r, not on the magnitudes of the moduli. To a residue tuple we attach the graph in which vertices i and j are adjacent precisely when aᵢ ≡ aⱼ (mod gcd(nᵢ,nⱼ)). Inclusion–exclusion makes the covered density a clique-weighted function of this graph. For every forced edge set, a finite-abelian-group kernel calculation counts compatible tuples by a Smith-normal-form lattice index. Boolean Möbius inversion then counts the tuples with each exact graph. Maximizing over the graphs with positive count gives the exact extremal density. The resulting factoring-free algorithm uses 2^{O(r²)} poly(B) bit operations, where B is the binary input length. We also give an independent prime-power layer formula for the kernel counts, a reduction to gcd kernels, and a factorization over the connected components of the non-coprimality graph. The construction is compatible with known arithmetic-coloring and abelian-arrangement machinery; the contribution is its exact-stratum composition with the Erdős–Graham density objective.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "residue classes",
    "covering systems",
    "natural density",
    "Smith normal form",
    "Möbius inversion",
    "fixed-parameter algorithms",
    "EP-278",
    "open mathematics",
    "mathematical proof",
    "math.CO",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-02",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/ep-278/",
  "pdf_url": "https://eulersolve.org/papers/ep-278/paper.pdf?v=9c67d60d97fb",
  "doi": "10.5281/zenodo.22244392",
  "zenodo_record_url": "https://zenodo.org/records/22244392",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}