# P2878: Orders of ternary row-orthogonal matrices with a full row

- ID: `P2878`
- Reference: `ternary-orthogonal-full-row-order`
- Page: https://theoremdb.org/statements/P2878
- Record maturity: Reviewed problem with recorded work

## 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 \(4\)?

### Definitions

- **Definition.** Rows are pairwise orthogonal when their standard Euclidean dot product is zero for every two distinct rows.
- **Definition.** 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.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

A complete resolution must satisfy this condition: 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.

### Background and intake notes

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.

- Original intake status: 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.
- 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: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `ternary-orthogonal-full-row-order`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/446995/orthogonal-vectors-with-entries-from-1-0-1
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Justin D. Christian and Bryan L. Shader, “Nonexistence Results for Hadamard-Like Matrices,” Electronic Journal of Combinatorics 11 (2004), Note N1. abstract and Theorems 7 through 9 on forbidden orders p^k, 2p^k, and 3p https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1n1/pdf/
   - website; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>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. question 418009, checked through the Stack Exchange API on 2026-08-01 https://mathoverflow.net/questions/418009
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.