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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 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.)