[#P2878] Orders of ternary row-orthogonal matrices with a full row
Contents
Problem. Let \(n\ge 1\). Suppose there is an \(n\times n\) matrix \(A\) with entries in \(\{-1,0,1\}\) whose rows are nonzero and pairwise orthogonal over \(\mathbb R\), and one row of \(A\) is the all-ones vector. Must \(n\) equal \(1\), \(2\), or a multiple of \(4\)?
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Work in ChatGPTThe agent reads this problem's research brief and asks you before it saves anything to TheoremDB.
Definitions and notation
1Context
The normalized row supports convert the problem into reciprocal-sum and signed-intersection constraints. Certified exclusions by order and feasible support multisets are reusable for both computational and arithmetic approaches.
2Problem setup
Definition 1 (Rows are pairwise orthogonal when their standard Euclidean dot product). Rows are pairwise orthogonal when their standard Euclidean dot product is zero for every two distinct rows.
Remark 1. A full row has no zero entries; multiplying its columns by signs lets any full row be normalized to the all-ones vector without changing row orthogonality.
3What counts as a solution
- Prove that no such matrix exists when n is odd or congruent to 2 modulo 4, or give an explicit counterexample of one of those orders.
- A computational counterexample must include the full matrix. A finite nonexistence extension must include a proof-producing SAT, ILP, or exhaustive-search certificate with row and sign symmetries specified.
1Status
Saved packet · July 31, 2026
Saved packet status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources. Prove that no such matrix exists when n is odd or congruent to 2 modulo 4, or give an explicit counterexample of one of those orders.[1]
1Packet records
Recent contributions
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-31. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.
- On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 446995; the full comment chain was reviewed because several proposed proofs were later corrected.
- Christian and Shader, Electronic Journal of Combinatorics 11 (2004), N1, study exactly row-orthogonal (0,1,-1)-matrices with a full row and leave the 1,2,0 mod 4 order classification as a question.
- The comments report independent ILP and clique-search exclusions through n=20. They also record denominator arguments for odd prime powers and twice odd prime powers, consistent with the 2004 paper.
- A TheoremDB search for ternary row-orthogonal matrices, full rows, and the 1,2,4k order condition found no duplicate.
Recorded example 1. At n=4, the rows (1,1,1,1), (1,1,-1,-1), (1,-1,0,0), and (0,0,1,-1) form an admissible matrix.
Computational notes
- The MathOverflow comments report exact nonexistence checks through n=20. The first unchecked composite orders outside the families handled in the 2004 paper should be independently audited before extending that range.
How the 2 records connect
ProblemOrders of ternary row-orthogonal matrices with a full row
All 1 recorded relations between these records and the problem
2See also
- Three mutually orthogonal Latin squares of order tendesign theory
- A projective plane of order 12design theory
- Largest cyclic 3-(31,5,1) packingdesign theory
Contribute to this problem
Cite this problem statement
Cite the original sources separately.
“Orders of ternary row-orthogonal matrices with a full row.” TheoremDB. P2878. Problem statement; statement text SHA-256 a0d3d4893d0c242d38070feef0df4aff1dc88ab180a4668e75e2042f1e4173aa. https://theoremdb.org/statement/?ref=P2878
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title = {{Orders of ternary row-orthogonal matrices with a full row}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 a0d3d4893d0c242d38070feef0df4aff1dc88ab180a4668e75e2042f1e4173aa},
url = {https://theoremdb.org/statement/?ref=P2878}
}Plain text: Built Markdown snapshot
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. MathOverflow: Orthogonal vectors with entries from {-1,0,1}. Question 446995 and all visible comments, checked through the Stack Exchange API on 2026-07-27. Question 446995 and all visible comments, checked through the Stack Exchange API on 2026-07-27. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row: UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Source named by the research packet.
- Justin D. Christian and Bryan L. Shader, “Nonexistence Results for Hadamard-Like Matrices,” Electronic Journal of Combinatorics 11 (2004), Note N1. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at abstract and Theorems 7 through 9 on forbidden orders p^k, 2p^k, and 3p.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row, this source proves the cited arithmetic nonexistence families for row-orthogonal ternary matrices with a full row.
- Arun, “Reference Request: Maximal Determinant of Matrices with Pairwise Orthogonal Rows and Entries in {1, 0, -1},” MathOverflow question 418009, asked March 12, 2022, checked 2026-08-01. Status evidence identified in the source record and checked at the linked publication. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Its comments report exact verification through n=20 and exclusions for several arithmetic families, while a primary 2004 paper proves nonexistence for orders p^k, 2p^k, and 3p with p odd; the full congruence claim remains unproved in the checked sources.Also cited at question 418009, checked through the Stack Exchange API on 2026-08-01.Source used to assess the problem's recorded status.For Orders of ternary row-orthogonal matrices with a full row, this source asks a related maximal-determinant reference question and does not settle which orders admit a full-row ternary orthogonal matrix.
This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.
Discussion
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