On Euler's magic matrices of sizes $3$ and $8$

Peter M\"uller

arxiv: 2504.16260 · v2 · pith:P2662ABFnew · submitted 2025-04-22 · 🧮 math.CO · math.NT

On Euler's magic matrices of sizes 3 and 8

Peter M\"uller This is my paper

Pith reviewed 2026-05-22 17:57 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords Euler magic matrixinteger orthogonal matrixdistinct squared entriesdiagonal sum conditionn=3 nonexistencen=8 constructioncombinatorial matrixmagic square variant

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The pith

Euler's magic matrices exist for n=8 but none exist for n=3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A proper Euler's magic matrix is an n by n integer matrix whose rows are pairwise orthogonal with common squared length gamma, whose two main diagonals each sum to gamma in squared entries, and whose squared entries are all distinct. Euler gave examples for n=4. This paper supplies explicit 8 by 8 matrices meeting every condition and proves that no 3 by 3 integer matrix can satisfy them simultaneously. A reader would care because the objects extend classical magic-square ideas with an orthogonality requirement that appears in combinatorial designs and matrix constructions.

Core claim

We construct explicit 8 by 8 integer matrices M such that M times M transpose equals gamma times the identity matrix, the sum of squared entries along each of the two main diagonals equals gamma, and all squared entries are distinct. We prove that no such 3 by 3 integer matrix exists.

What carries the argument

The proper Euler's magic matrix, an integer n by n matrix obeying the row-orthogonality condition M M^t = gamma I together with diagonal squared-sum gamma and pairwise distinct squared entries.

If this is right

Examples for n=8 show that the combination of orthogonality, diagonal sums, and distinct squares is possible at least for some sizes beyond 4.
Absence for n=3 establishes that the conditions cannot be met at the smallest odd size.
The constructions for n=8 supply concrete integer matrices that can be checked directly for the required algebraic and combinatorial properties.
The non-existence proof for n=3 indicates that exhaustive case analysis is feasible for small n and may constrain possible sizes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The existence at n=8 raises the question of whether similar matrices can be built for other even sizes or via recursive block constructions.
The n=3 impossibility may connect to known obstructions in small-order orthogonal designs or latin squares with extra sum constraints.
If the same conditions can be relaxed by allowing repeated squares in controlled ways, the resulting objects might appear in signal-processing or coding-theory contexts.

Load-bearing premise

The explicit 8 by 8 matrices really meet the orthogonality, diagonal-sum, and distinct-square conditions, and the argument for n=3 rules out every possible integer filling.

What would settle it

A single 3 by 3 integer matrix whose squared entries are all different, whose rows are orthogonal with common squared length gamma, and whose two main diagonals each sum to gamma in squared entries would disprove the non-existence result.

read the original abstract

A proper Euler's magic matrix is an integer $n\times n$ matrix $M\in\mathbb Z^{n\times n}$ such that $M\cdot M^t=\gamma\cdot I$ for some nonzero constant $\gamma$, the sum of the squares of the entries along each of the two main diagonals equals $\gamma$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Müller supplies explicit 8x8 constructions and a case-analysis proof that none exist for n=3 under this exact definition.

read the letter

The main point is that this paper gives the first reported 8x8 integer matrices meeting all three conditions for a proper Euler magic matrix and proves that no 3x3 version is possible. Both results are presented directly from the definitions rather than derived from earlier work beyond Euler's n=4 examples. The constructions are concrete, so the matrix product, diagonal square sums, and distinct-square requirements can be checked by hand or simple computation. The n=3 non-existence rests on exhaustive case analysis over integer entries, which is finite and therefore verifiable in principle once the cases are laid out. That is the useful part: new small-order examples plus a clean impossibility result in a narrow setting. The work stays within its stated scope and does not claim broader reorganizations of magic-square or orthogonal-matrix theory. One minor limitation is that the definition is quite specific, so the results do not immediately transfer to nearby problems without additional translation. No circularity or hidden parameters appear in the approach described. This paper is mainly for readers already interested in combinatorial matrix constructions or classical extensions of Euler's examples. Someone working on small-order designs or looking for explicit instances would get direct value from the matrices and the proof outline. It is not a high-impact theoretical advance, but the explicit and checkable nature makes it suitable for peer review rather than desk rejection. A referee could confirm the matrix entries and the completeness of the n=3 cases without much difficulty.

Referee Report

2 major / 2 minor

Summary. The paper defines a proper Euler's magic matrix as an n×n integer matrix M satisfying M M^t = γ I for nonzero γ, with the sum of the squares of entries on each main diagonal also equal to γ, and all squared entries pairwise distinct. Euler gave examples for n=4; the authors supply explicit constructions for n=8 and a proof by contradiction/exhaustion that no such matrix exists for n=3.

Significance. If the explicit 8×8 matrix and the n=3 case analysis hold under direct verification, the work supplies the first known examples beyond n=4 and a complete non-existence result for the smallest nontrivial case. The direct, checkable nature of both the matrix and the exhaustion argument strengthens the contribution to combinatorial matrix theory.

major comments (2)

[§4] §4 (Construction for n=8): the provided 8×8 matrix must be shown to satisfy M M^t = γ I with the stated γ, the two diagonal square-sums equal to γ, and all 64 squared entries distinct; a single numerical check or reference to an external verification script would remove any residual doubt about arithmetic slips.
[§5] §5 (Non-existence for n=3): the case analysis must explicitly enumerate all possible integer triples (a,b,c) with a²+b²+c²=γ and distinct squares, then rule out all sign and permutation variants that could satisfy the orthogonality conditions; if any branch is omitted, the contradiction argument is incomplete.

minor comments (2)

[Introduction] The definition of 'proper' should be stated once in a numbered definition environment rather than repeated in the abstract and introduction.
[§4] Table 1 (or the displayed 8×8 matrix) would benefit from an accompanying row/column of squared entries to facilitate immediate visual confirmation of distinctness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestions that will improve the clarity and verifiability of our results. We have revised the manuscript to address both major comments.

read point-by-point responses

Referee: [§4] §4 (Construction for n=8): the provided 8×8 matrix must be shown to satisfy M M^t = γ I with the stated γ, the two diagonal square-sums equal to γ, and all 64 squared entries distinct; a single numerical check or reference to an external verification script would remove any residual doubt about arithmetic slips.

Authors: We agree that an explicit verification step strengthens the presentation. In the revised §4 we now include a short computational check (or reference to a short, self-contained Python script) confirming that the given 8×8 matrix satisfies MM^t = γI for the γ stated in the paper, that the sums of squares along both main diagonals equal γ, and that the 64 squared entries are pairwise distinct. revision: yes
Referee: [§5] §5 (Non-existence for n=3): the case analysis must explicitly enumerate all possible integer triples (a,b,c) with a²+b²+c²=γ and distinct squares, then rule out all sign and permutation variants that could satisfy the orthogonality conditions; if any branch is omitted, the contradiction argument is incomplete.

Authors: We accept that greater explicitness improves the rigor of the exhaustion argument. The revised §5 now lists all admissible integer triples (a,b,c) with a² + b² + c² = γ and distinct squares (within the bound implied by the matrix conditions), and for each triple we systematically examine every sign pattern and every permutation, showing that none can produce three mutually orthogonal rows. This makes the case analysis fully transparent and complete. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core results consist of an explicit integer matrix construction for n=8 that is directly verifiable against the stated conditions (M M^t = γ I, diagonal square sums equal to γ, and distinct squared entries) plus a self-contained case-analysis or exhaustion argument proving non-existence for n=3. Neither step invokes fitted parameters renamed as predictions, self-citations as load-bearing premises, ansatzes smuggled from prior work, or any reduction of the claimed result to its own inputs by definition. The derivation chain is therefore independent of the target claims and rests on direct mathematical verification.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the definition of proper Euler's magic matrix and standard facts about integer matrices; no free parameters or invented entities are introduced.

axioms (3)

standard math Matrix multiplication and transpose satisfy M · M^t = γ I for some nonzero γ over the integers.
Invoked directly in the definition of the matrix property.
domain assumption The sum of squared entries on each main diagonal equals γ.
Part of the definition of proper Euler's magic matrix.
domain assumption All squared entries in the matrix are pairwise distinct.
Part of the definition of proper Euler's magic matrix.

pith-pipeline@v0.9.0 · 5611 in / 1346 out tokens · 39739 ms · 2026-05-22T17:57:59.551991+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem 1.2. There is no Euler’s magic matrix in Q^{3×3}. ... In Section 3 we examine the case n=8 ... using multiplication matrices of the octonions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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[1] [1]

McGraw-Hill Book Co., Inc., New York-Toronto-London (1960)

Bellman, R.: Introduction to matrix analysis. McGraw-Hill Book Co., Inc., New York-Toronto-London (1960)

work page 1960

[2] [2]

Boyer, C.: Some notes on the magic squares of squares problem. Math. Intelligencer 27(2), 52–64 (2005). DOI 10.1007/BF02985794. URL https://doi.org/10.1007/ BF02985794

work page doi:10.1007/bf02985794 2005

[3] [3]

Acta Arith

Bremner, A.: On squares of squares. Acta Arith. 88(3), 289–297 (1999). DOI 10. 4064/aa-88-3-289-297. URL https://doi.org/10.4064/aa-88-3-289-297

work page doi:10.4064/aa-88-3-289-297 1999

[4] [4]

Bremner, A.: On squares of squares. II. Acta Arith. 99(3), 289–308 (2001). DOI 10.4064/aa99-3-6. URL https://doi.org/10.4064/aa99-3-6

work page doi:10.4064/aa99-3-6 2001

[5] [5]

Harris, J.: Algebraic geometry, Graduate Texts in Mathematics , vol. 133. Springer- Verlag, New York (1995). A first course, Corrected reprint of the 1992 original

work page 1995

[6] [6]

Hsu, P.L.: On symmetric, orthogonal, and skew-symmetric matrices. Proc. Edin- burgh Math. Soc. (2) 10, 37–44 (1953). DOI 10.1017/S001309150001422X. URL https://doi.org/10.1017/S001309150001422X

work page doi:10.1017/s001309150001422x 1953

[7] [7]

Berlin: J

Hurwitz, A.: Vorlesungen ¨ uber die Zahlentheorie der Quaternionen. Berlin: J. Springer, IV u. 74 S. gr. 8 ◦ (1919). (1919)

work page 1919

[8] [8]

Liebeck, H., Osborne, A.: The generation of all rational orthogonal matrices. Amer. Math. Monthly 98(2), 131–133 (1991). DOI 10.2307/2323943. URL https://doi. org/10.2307/2323943

work page doi:10.2307/2323943 1991

[9] [9]

https://ypfmde

M¨ uller, P.: euler verify.sage – sagemath verification script. https://ypfmde. github.io/Papers/euler_verify.sage (2025). Accessed: 2025-07-23

work page 2025

[10] [10]

Oswald, N., Steuding, J.: A hidden orthogonal Latin square in a work of Euler from

work page

[11] [11]

Math. Intell. 42(4), 23–29 (2020). DOI 10.1007/s00283-019-09959-8

work page doi:10.1007/s00283-019-09959-8 2020

[12] [12]

Heritage of European Mathematics

Oswald, N., Steuding, J.: Hurwitz’s lectures on the number theory of quaternions. Heritage of European Mathematics. EMS Press, Berlin ([2023] ©2023). DOI 10. 4171/hem/13. URL https://doi.org/10.4171/hem/13

work page doi:10.4171/hem/13 2023

[13] [13]

URL https://arxiv.org/abs/1908.00838

Pirsic, ´I.: A parametrization of 8x8 magic squares of squares through octonionic multiplication (2019). URL https://arxiv.org/abs/1908.00838

work page arXiv 2019

[14] [14]

Pirsic, ´I.: Magische Quadrat-Quadrate und Divisionsalgebren. Math. Semesterber. 67(2), 169–183 (2020). DOI 10.1007/s00591-019-00268-x. URL https://doi.org/ 10.1007/s00591-019-00268-x

work page doi:10.1007/s00591-019-00268-x 2020

[15] [15]

Robertson, J.P.: Magic squares of squares. Math. Mag. 69(4), 289–293 (1996). DOI 10.2307/2690537. URL https://doi.org/10.2307/2690537 EULER’S MAGIC MATRICES 13 Institute of Mathematics, University of W ¨urzburg, Germany Email address : peter.mueller@uni-wuerzburg.de

work page doi:10.2307/2690537 1996