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A308838 Orders of Parker finite fields of odd characteristic. 1
3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 31, 43, 47, 67, 243 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..16.

Onno M. Cain, Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.

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).

EXAMPLE

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 from 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.

PROG

(SageMath)

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):

    if q % 2 == 0: continue

    if len(factor(q)) > 1: continue

    print q, msos_search(GF(q), single=True)

CROSSREFS

Sequence in context: A098903 A061345 A238266 * A080429 A326581 A050150

Adjacent sequences:  A308835 A308836 A308837 * A308839 A308840 A308841

KEYWORD

nonn,hard,more

AUTHOR

Onno M. Cain, Jun 27 2019

STATUS

approved

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Last modified January 28 22:40 EST 2020. Contains 331328 sequences. (Running on oeis4.)