A364264 -id:A364264 - cp's OEIS frontend

A308838 Orders of Parker finite fields of odd characteristic.

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.

A348263 Orders of Parker fields.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 31, 32, 43, 47, 64, 67, 128, 243, 256, 512, 1024, 2048, 4096

Offset: 1

Thomas Scheuerle, Oct 09 2021

If a traditional magic square of squares does not exist with elements from a field F, then F is said to be a Parker field.
It is conjectured that these are the only such fields.
Appears to be essentially the same as A308838. - R. J. Mathar, Oct 15 2021
It appears that there is a mistake in the paragraph after Conjecture 7.2 of the Cain article. It claims that there are only 17 finite Parker fields, although Lemma 5.2 clearly shows that all fields of order 2^k are Parker. I think the corrected conjecture should state that there are only 16 finite Parker fields of odd order. - Yevhenii Diomidov, Jan 19 2022

			The field GF(29), for example, is not Parker since:
  ----------------
  |9^2 |11^2|1^2 | mod 29 = 0
  ----------------
  |6^2 |0^2 |14^2| mod 29 = 0
  ----------------
  |12^2|16^2|8^2 | mod 29 = 0,
  ----------------
with the same property for columns and main diagonals.

O. M. Cain, Gaussian Integers, Rings, Finite Fields,and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Matt Parker, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).

Cf. A308838, A309810, A364264.

Missing even terms added by Yevhenii Diomidov, Jan 19 2022

A380966 a(n) is an upper bound such that there exists an m X m magic square of n-th powers for all m >= a(n).

Original entry on oeis.org

36, 52, 84, 140, 164, 196, 224, 252, 284, 312, 344, 372, 404, 436, 468, 500, 532, 564, 596, 632, 664, 696, 732, 764, 796, 832, 864, 900, 936, 968, 1004, 1036, 1072, 1108, 1144, 1180, 1212, 1248, 1284, 1320, 1356, 1392, 1428, 1464, 1500, 1536, 1572, 1608, 1644, 1680

Offset: 2

Paolo Xausa, Feb 09 2025

nonn

See Rome and Yamagishi (2024), eq. (2.2).

In particular, an m X m magic square of squares is proved to exist for all m >= 36. Combined with previous results that show the existence of such squares for 4 <= m <= 64, it follows that an m X m magic square of squares exists for all m >= 4. The 3 x 3 case is still unsolved.

Brady Haran and Matt Parker, A Magic Square Breakthrough, YouTube Numberphile video, 2025.
Nick Rome and Shuntaro Yamagishi, On the existence of magic squares of powers, arXiv:2406.09364v2 [math.NT], 2024.
Index entries for sequences related to magic squares

Cf. A364264.

A380966[n_] := 20 + 4*If[2 <= n <= 4, 2^n, Ceiling[n*(Log[n] + 4.20032)]];
Array[A380966, 50, 2]

a(n) = 4*2^n + 20, if 2 <= n <= 4;

a(n) = 4*ceiling(n*(log(n) + 4.20032)) + 20, if n >= 5. Cf. Rome and Yamagishi (2024), eq. (2.2).

A379179 The Anti-Parker Square, read by rows.

Original entry on oeis.org

8288641, 16492609, 4515625, 5992609, 9765625, 13538641, 15015625, 3038641, 11242609

Offset: 1

Paolo Xausa, Dec 17 2024

This square has been discovered by Ayliean MacDonald (see links) while attempting to create a magic square of squares.
It's an example of a 3 X 3 square where all rows, columns and diagonals sum to the same number (in this case 29296875), but only the entries in the diagonals are squares.
The Parker Square (A364264) was an attempt (which failed in a different way) to solve the same problem.

			The Anti-Parker Square is:
  [  8288641 16492609  4515625 ]
  [  5992609  9765625 13538641 ]
  [ 15015625  3038641 11242609 ]
Or equivalently:
  [  2879^2  16492609   2125^2 ]
  [ 5992609    3125^2 13538641 ]
  [  3875^2   3038641   3353^2 ]

Brady Haran and Ayliean MacDonald, The Anti-Parker Square, YouTube Numberphile video, 2024.
Brady Haran and Matt Parker, The Parker Square, YouTube Numberphile video, 2016.

Cf. A364264.

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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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A348263 Orders of Parker fields.

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A380966 a(n) is an upper bound such that there exists an m X m magic square of n-th powers for all m >= a(n).

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A379179 The Anti-Parker Square, read by rows.

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