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A parametrization of 8x8 magic squares of squares through octonionic multiplication

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The rows of the symbolic octonion product $A(e_iP)$ form an 8x8 semi-magic square of squares with orthogonal rows and constant $N(A)N(P)$.

desk verdict A clean, honest octonionic extension of Euler/Hurwitz that is worth refereeing once the load-bearing distinctness check is made reproducible. read the letter →

arxiv 1908.00838 v2 pith:QBRBAPPW submitted 2019-08-02 math.HO math.CO

Ísabel Pirsic This is my paper · ORCID
classification math.HOmath.CO MSC 05B1517A75
keywords magicsquaresofoctonionssemi-magicsquareorthogonalrowsCayley-Dicksonconstructionnorm-multiplicativealgebrasSO(8)composition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the classical 4x4 quaternionic parametrization of squares of squares to dimension 8. For symbolic octonions $A$ and $P$, it forms the 8x8 matrix whose $i$-th row is the component vector of $A(e_iP)$, where $e_i$ runs over the standard octonionic basis, and proves that this matrix has orthogonal rows, entries that are integer polynomials in the 16 variables, and constant row-sum $N(A)N(P)$. This yields explicit integral 8x8 semi-magic squares of squares whenever the variables are evaluated at rational points with distinct entries; one example shown has magic constant 9476. The argument also explains why the same construction cannot be repeated for the 16-dimensional sedenions: their multiplication is not norm-multiplicative, so the row-orthogonality mechanism breaks.

What carries the argument

The load-bearing object is the square matrix generated by the eight products $A(e_iP)$. The proof mechanism is the norm identity $N(Ae_iP)=N(A)N(e_i)N(P)=N(A)N(P)$, which holds because octonionic multiplication is norm-multiplicative, combined with the fact that left and right multiplication by a fixed unit octonion are orthogonal transformations of the eight-dimensional space. These two properties force the eight row vectors to be pairwise orthogonal and of equal length, so their component-squared sums are all the same. The basis multiplication table supplies the $\pm1$ coefficients, and the Cayley-Dickson convention for the octonionic basis fixes the explicit matrix.

What would settle it

Feed the symbolic matrix $M$ defined by $A(e_iP)$ into any computer algebra system and compute $MM^\top$; if any off-diagonal entry is a nonzero polynomial, Theorem 1 is false. For the claimed non-extension, compute the analogous 16x16 matrix for sedenions and check whether $MM^\top$ has zero off-diagonal entries; a nonzero off-diagonal polynomial there would confirm the paper's negative claim.

Watch

Extended reading notes

Core claim

The central discovery, stated as Theorem 1, is that for any symbolic octonions $A = ae_1+\cdots+he_8$ and $P = pe_1+\cdots+we_8$, the matrix $M$ with row vectors $\mathrm{comp}(A(e_iP))$ is a semi-magic square of squares: every entry is a square of an integer-linear polynomial, the rows are mutually orthogonal, and the common value of the row sums of squares is $N(A)N(P)$. The proof uses the fact that multiplication by a fixed octonion on the left or right preserves orthogonality and scales every length by the same factor, together with the octonionic basis table, which gives coefficients $\pm1$. Evaluated at rational vectors with all entries distinct, the construction produces integral semi-magic squares; the paper gives a concrete matrix with constant 9476 and a parameter pair yielding all absolute values distinct with constant 43617.

Load-bearing premise

The construction collapses if left or right octonionic multiplication fails to preserve the angles between basis vectors while scaling all lengths by one common factor; that single algebraic property supports the row orthogonality and the constant row sum.

Editorial extensions

If this is right

  • Any rational evaluation with distinct entries gives an integral 8x8 semi-magic square of squares with orthogonal rows; the paper exhibits one with constant 9476.
  • The parametrization is symbolically explicit in 16 variables, so it provides a whole family rather than isolated examples, and the row-sum constant always factors as $N(A)N(P)$.
  • The diagonal (fully magic) case is not covered: the paper reports that permuting rows to force diagonal sums does not lead to side conditions as in the 4x4 case, and a small random search found no fully magic example.
  • The construction cannot be pushed to dimension 16 in the Cayley-Dickson series, because the sedenions lack norm multiplicativity; in other composition algebras a modified orthogonality condition, $MQM^\top$ instead of $MM^\top$, would be needed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is to search the 16-parameter lattice systematically, including the half-integer octonion points the paper flags, to see whether a fully magic 8x8 square exists; the current small random search does not rule it out.
  • The orthogonal-row structure means the matrix is an orbit of a scaled $SO(8)$ action on the basis; this geometric view might make the diagonal constraint a condition on the permutation of that orbit, potentially explaining why the diagonal equations are harder.
  • The same row recipe in other composition algebras could yield rational semi-magic squares with a different orthogonality notion, but the interpretative meaning of 'magic' would change accordingly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper presents an octonionic analogue of Euler's quaternionic parametrization of 4x4 magic squares of squares. For symbolic octonions A and P, the rows of the 8x8 matrix M are defined as the component vectors of A(e_i P) for the octonionic basis elements e_i. Theorem 1 asserts that M is a symbolic semi-magic square of squares with entries in Z[a,...,h,p,...,w], with pairwise orthogonal rows and constant row squared norm N(A)N(P). The proof relies on the standard fact that left and right octonionic multiplication act as scaled orthogonal maps, citing Conway--Smith. The paper gives an explicit integral example with constant 9476 and reports an unsuccessful randomized search for a fully magic (diagonal) solution. The final paragraph argues that the analogous sedenionic construction cannot work because norm multiplicativity fails.

Significance. If the distinctness claim is properly substantiated, the result is a clean 8-dimensional generalization of Euler's construction, obtained without fitted parameters and with falsifiable explicit examples. The algebraic core---row orthogonality and constant norm---follows from standard octonionic norm multiplicativity as cited from Conway--Smith, and the paper correctly identifies the octonion-based construction as the natural stopping point in the Cayley--Dickson series. The main weakness is that the proof delegates the crucial verification that all 64 symbolic entries are distinct to an unavailable Sage computation: the scripts are not shipped, and the multiplication table is specified only as the Cayley--Dickson extension as implemented in Sage. The result is not fully checkable as it stands, and this gap is load-bearing for the claimed parametrization of genuine semi-magic squares.

major comments (1)
  1. [Proof of Theorem 1 (paragraph beginning 'That all entries are different…')] The assertion 'That all entries are different can be verified by calculation, e.g., using Sage [5]' is load-bearing: the intended parametrization yields a (semi-)magic square only when the 64 entries are distinct, as the later examples emphasize. The manuscript provides neither the Sage script, nor the explicit multiplication table used, nor a human-checkable argument. The statement that scripts are available from the author by request is not sufficient for a refereed journal. Please either include the verification code and its output, or replace the assertion with a proof (for example, by exhibiting a monomial that occurs in exactly one of the 64 polynomial entries, or by providing two evaluations that force polynomial distinctness).
minor comments (5)
  1. [Final paragraph (sedenions)] The statement that because norm multiplicativity fails for sedenions, 'consequently the analog matrix does not have orthogonal rows' is an overstatement. Failure of norm multiplicativity shows that the octonionic proof does not carry over, but it does not logically rule out orthogonal rows for special parameter choices. Please either prove the stronger claim or revise it to say that the same argument does not apply.
  2. [Theorem 1 and surrounding text] The term 'symbolic semi-magic square of squares' is not defined. In particular, clarify whether distinctness of all 64 entries is part of the definition; if so, make it explicit in the statement of Theorem 1 rather than only in the proof.
  3. [References and typos] Reference [2] has 'nouvelle espéce' (should be 'espèce'), reference [3] has 'academlae' (should be 'academiae'), and the text has 'enlightning' (should be 'enlightening').
  4. [Example section] The displayed 8x8 matrix with constant 9476 is not explicitly linked to the later parameters A = (8,-2,-4,8,-4,-1,-5,-4) and P = 1/2(5,7,-1,-3,-7,1,7,1). Please state explicitly that this matrix is the evaluation of the symbolic matrix at those parameters, or show the evaluation.
  5. [Proof of Theorem 1, citation [1]] The proof cites Conway--Smith for the fact that multiplication with unit octonions models SO(8). Please provide a precise chapter, section, or theorem number to help the reader verify this standard but nontrivial fact.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the octonionic parametrization is derived from standard algebraic facts, not from its own conclusion.

full rationale

The paper's central claim, Theorem 1, is that the rows of M formed from components of A(e_i P) are orthogonal and have constant norm N(A)N(P). This follows directly from standard octonionic facts: norm multiplicativity and the fact that left and right multiplication by fixed octonions act as scaled orthogonal maps. The paper explicitly cites Conway and Smith for the SO(8) property, which is an external, well-established mathematical source, not a self-citation and not a premise that depends on the theorem being proved. No parameter is fitted to any target matrix, and the theorem does not predict a quantity that was used to define the construction. The only weak point is the statement that distinctness of all entries 'can be verified by calculation, e.g., using Sage', with the scripts not shipped; this is an omitted verification or reproducibility gap, not a circularity, because it neither assumes the conclusion nor defines the output in terms of a fitted input. The negative result for sedenions is justified by Hurwitz's theorem on normed division algebras, again an external standard fact. Overall, the derivation chain is self-contained given established octonionic algebra, and no step reduces to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters enter the theorem; the symbolic coordinates a,...,w are the intended variables of the parametrization, not ad hoc constants. The proof relies on standard octonion facts and a stated Sage convention, and introduces no new entities.

assumptions (4)
  • standard math Octonionic norm is multiplicative: N(xy)=N(x)N(y) for all octonions x,y.
    Used to show each row of M has equal squared sum N(A)N(P), in the discussion after the four-squares identity and in the proof of Theorem 1.
  • standard math Left and right multiplication by fixed octonions are scaled orthogonal maps; unit octonion multiplication models SO(8).
    Cited to Conway-Smith [1] and invoked in the proof: 'multiplication with unit octonions models SO(8)'.
  • standard math No normed real division algebra exists beyond dimension 8 (Hurwitz theorem); the sedenion norm is not multiplicative.
    Supports the claim that the construction cannot be extended to 16 dimensions; the paper cites this via [1].
  • domain assumption The octonion multiplication table used is the Cayley-Dickson extension implemented in Sage, which fixes the sign pattern and distinctness check.
    The paper states 'we chose the Cayley-Dickson extension ... as given by Sage'; the exact matrix entries and the all-distinct verification depend on this convention.

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Cite this review

Pith. "Pith review of A parametrization of 8x8 magic squares of squares through octonionic multiplication." pith.science (2026). https://pith.science/paper/QBRBAPPW

@misc{pith2026190800838,
  author       = {Pith},
  title        = {Pith review of: A parametrization of 8x8 magic squares of squares through octonionic multiplication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBRBAPPW}},
  note         = {Machine review of arXiv:1908.00838}
}
read the original abstract

In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be carried even further in the Cayley-Dickson construction series.

Figures

Figures reproduced from arXiv: 1908.00838 by the authors.

Figure 1
Figure 1. The quaternionic magic square of squares [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The octonionic semimagic square of squares [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A proper Euler magic matrix of order $5$

    math.GM 2026-07 accept novelty 6.0 of 10

    An explicit 5×5 integer matrix is constructed whose transpose-times-self is scalar, whose row/column and both diagonal square-sums are that scalar, and whose 25 squared entries are all distinct.

Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages · cited by 1 Pith paper

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    A.Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Springer, Berlin, 1919

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    Reprinted in Opera Omnia, I.7, 291-392

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    L.Euler, Problema algebraicum ob affectiones prorsus singulares i memorabile, Novi commentarii academlae scientiarum Petropolitanae, 15 (1770) 1771, 75-106. Reprinted in Opera Omnia, I-6, 287-315

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