bibkey: wu2025pyramidalcomplement authors: Chai Wah Wu year: 2025 title: Algorithms for Complementary Sequences doi: 10.5281/zenodo.17535229 url: https://math.colgate.edu/~integers/z95/z95.pdf claim: “Conjecture 1. For k >= 9, (6) holds for all n >= 1.” strata_touched:

• D5/S3/Arith/WuPyramidalComplement license: citation-only triage: anchor

Algorithms for Complementary Sequences

Wu defines the complementary sequence as the increasing sequence of positive integers absent from a prescribed increasing sequence. On printed page 2 this is the n-th positive integer outside the source image, so the index is one-based and zero is not a complement term.

For the k-gonal-pyramidal sequence, printed page 11 gives Equation (6). With

h = floor((6n/(k-2))^(1/3)),
U = (k-2)h^3 + 3(k-1)h^2 + (2k-1)h + 6,
L = h(h-1)(h(k-2)+k+1),

the proposed term is n+h+1 when 6n >= U, n+h-1 when the first branch fails and 6n <= L, and n+h otherwise. Printed page 12 states Conjecture 1:

For k >= 9, (6) holds for all n >= 1.

The earlier results in the article prove formulas for other parameter ranges or for sufficiently large indices; they do not prove this all-index statement.

The checked source is the 26-page INTEGERS 25 (2025), article A95 PDF. Its SHA-256 is b4c7db146554cb2b4e88901ab4fd3799a41f46797b9b9d4e851f39aca60c690b.

Verified locator

• DOI: https://doi.org/10.5281/zenodo.17535229
• URL: https://math.colgate.edu/~integers/z95/z95.pdf
• Exact source locations: complementary-sequence definition on printed page 2; Equation (6) on printed page 11; Conjecture 1 on printed page 12.

Literature boundary

The arXiv record 2409.05844 through version 10, the journal version, Wu’s publication list including its 2026 entries, and the author’s public sequence repository were checked within the stated bounded search. GitHub code searches for 2409.05844 and complementary sequences returned zero complete hits; the two pyramidal hits were a partial sequence definition and a formula registry, not this theorem. DataCite returned no indexed citations for the arXiv or journal record. Later-title searches returned papers about signal or Golay complementary sequences, not this number-theoretic conjecture.

OpenAlex and Semantic Scholar returned HTTP 429, so their citation graphs were not verified. The search is not exhaustive and establishes neither worldwide novelty nor priority. An earlier complete published resolution would supersede the open-problem assessment without affecting the formal theorem’s kernel validity.

Repository settlement

The repository proves the full source statement for every k >= 9 and n >= 1 as D5/S3/Arith/WuPyramidalComplement.wu_conjecture_one. The theorem’s frozen sourceStatementId is sha256:7b0c4f46be19f41459abfe30393dba3916c440231d0e079d0e55b3cd949dcf89; its containing module is frozen at sha256:627b09b5bbcd81cb4f97b0496c330457ceb734fb6810d90291ff360ba64cb6d7. This formal settlement does not strengthen the bounded novelty or priority claim above.