A348263 -id:A348263 - cp's OEIS frontend

A364264 The Parker Square, read by rows.

841, 1, 2209, 1681, 1369, 1, 529, 1681, 841

Offset: 1

Paolo Xausa, Jul 17 2023

Named after Matt Parker, who attempted (and failed) to create a 3 X 3 magic square of squares (still an open problem). The sum of entries in the rows, columns and one diagonal is 3051, but in the other diagonal the sum is 4107. Moreover, three entries are repeated (1^2, 29^2 and 41^2).

Cain (2019) cites this trivial semimagic square and calls a finite field a Parker field if no 3 X 3 magic square of squares can be constructed using 9 distinct squared elements.

			The Parker Square is:
  [  841    1 2209 ]
  [ 1681 1369    1 ]
  [  529 1681  841 ]
Or equivalently:
  [ 29^2  1^2 47^2 ]
  [ 41^2 37^2  1^2 ]
  [ 23^2 41^2 29^2 ]

Matt Parker, Humble Pi: A Comedy of Maths Errors, Penguin Books, UK, 2020, p. 6.

Onno M. Cain, Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Brady Haran and Matt Parker, The Parker Square, YouTube Numberphile video, 2016.
Brady Haran and Matt Parker, Finite Fields & Return of The Parker Square, YouTube Numberphile video, 2021.
Parker Square, The Parker Square on X (ex Twitter).
Wikipedia, Parker Square.
Christian Wolird, A New Transformation of the Magic Square of Squares, arXiv:2310.12164 [math.HO], 2023.
Index entries for sequences related to magic squares

Cf. A308838, A309810, A348263, A379179.

A308838 Orders of Parker finite fields of odd characteristic.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 31, 43, 47, 67, 243

Offset: 1

Onno M. Cain, Jun 27 2019

A field or ring is called "Parker" if no 3 X 3 magic square of 9 distinct squared elements can be formed. Conjecture: the sequence is complete.
Example: the fact that p=31 is listed is taken to mean one cannot construct a 3 X 3 magic square of distinct squared elements of the finite field of order 31.
Cain shows that each of the entries on the list corresponds to a Parker field and claims to have checked computationally that no other primes p < 1000 are on the list.
Labruna shows at least there are infinitely many primes not on the list.
The next term, if it exists, must be > 7500. - G. C. Greubel, Aug 16 2019

			Example: The prime p=29 does not appear in the sequence because one can in fact construct a 3 X 3 magic square of distinct squares over the finite field of order 29.
Construction:
   9^2 | 11^2 |  1^2
   6^2 |  0^2 | 14^2
  12^2 | 16^2 |  8^2
The square is valid evaluated mod 29 (example independently discovered by Woll and Cain). That is to say the entries of each row, column, and the two main diagonals sum to a multiple of 29.
Example: The fields corresponding to p^n = 3, 5, 7, 9, 11, and 13 are all Parker because each contains at most 7 distinct squared entries and cannot therefore provide the 9 distinct squares required for a magic square.

Onno M. Cain, Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Christian Woll, A Partial Residue Categorization of the Magic Square of Squares, arXiv:1809.03067 [math.NT], 2018.
Giancarlo Labruna, Magic Squares of Squares of Order Three Over Finite Fields, (2018). Theses, Dissertations and Culminating Projects. 138.
Matt Parker & Brady Haran, The Parker Square, Numberphile video (2016).

Cf. A309810, A348263, A364264.

def msos_search(F, single=False):
    squares = {x^2 for x in F}
    MSOS = []
    E = 0
    for A, I in Subsets(squares, 2):
        if A + I != 2*E: continue
        C, G = 1, -1
        B = 3*E - A - C
        D = 3*E - A - G
        F = 3*E - C - I
        H = 3*E - G - I
        if len(squares & {B,D,F,H}) < 4: continue
        if len({A,B,C,D,E,F,G,H,I}) < 9: continue
        if single: return [A,B,C,D,E,F,G,H,I]
        MSOS.append([A,B,C,D,E,F,G,H,I])
    E = 1
    sequences = []
    for A, I in Subsets(squares, 2):
        if A + I != 2*E: continue
        for C, G in sequences:
            B = 3*E - A - C
            D = 3*E - A - G
            F = 3*E - C - I
            H = 3*E - G - I
            if len(squares & {B,D,F,H}) < 4: continue
            if len({A,B,C,D,E,F,G,H,I}) < 9: continue
            if single: return [A,B,C,D,E,F,G,H,I]
            MSOS.append([A,B,C,D,E,F,G,H,I])
        sequences.append((A,I))
    return MSOS
for q in range(3, 500, 2):
    if len(factor(q)) > 1: continue
    print(q, msos_search(GF(q), single=True))

A309810 Orders of Parker rings.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 51, 52, 56, 57, 60, 62, 64, 66, 67, 68, 69, 72, 75, 76, 78, 80, 84, 86, 88, 92, 93, 94, 96, 100, 102, 104, 112, 114

Offset: 1

Michel Marcus, Aug 18 2019

nonn

A field or ring is called "Parker" if no 3 X 3 magic square of 9 distinct squared elements can be formed. Conjecture: the sequence is complete.

Example: the fact that p=31 is listed is taken to mean one cannot construct a 3 X 3 magic square of distinct squared elements of the ring of order 31.

Onno M. Cain, parker-ring-search SageMath code, Apr 24, 2019.
Onno M. Cain, Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Matt Parker & Brady Haran, The Parker Square, Numberphile video (2016).

Cf. A308838, A348263 (for finite fields), A364264.

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A364264 The Parker Square, read by rows.

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A308838 Orders of Parker finite fields of odd characteristic.

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A309810 Orders of Parker rings.

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