cp's OEIS Frontend

A308838 Orders of Parker finite fields of odd characteristic.

%I A308838 #60 Aug 07 2023 10:09:01
%S A308838 3,5,7,9,11,13,17,19,23,25,27,31,43,47,67,243
%N A308838 Orders of Parker finite fields of odd characteristic.
%C A308838 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.
%C A308838 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.
%C A308838 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.
%C A308838 Labruna shows at least there are infinitely many primes not on the list.
%C A308838 The next term, if it exists, must be > 7500. - _G. C. Greubel_, Aug 16 2019
%H A308838 Onno M. Cain, <a href="https://arxiv.org/abs/1908.03236">Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares</a>, arXiv:1908.03236 [math.RA], 2019.
%H A308838 Christian Woll, <a href="https://arxiv.org/abs/1809.03067">A Partial Residue Categorization of the Magic Square of Squares</a>, arXiv:1809.03067 [math.NT], 2018.
%H A308838 Giancarlo Labruna, <a href="https://digitalcommons.montclair.edu/etd/138">Magic Squares of Squares of Order Three Over Finite Fields</a>,  (2018). Theses, Dissertations and Culminating Projects. 138.
%H A308838 Matt Parker & Brady Haran, <a href="https://www.youtube.com/watch?v=aOT_bG-vWyg">The Parker Square</a>, Numberphile video (2016).
%e A308838 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.
%e A308838 Construction:
%e A308838    9^2 | 11^2 |  1^2
%e A308838    6^2 |  0^2 | 14^2
%e A308838   12^2 | 16^2 |  8^2
%e A308838 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.
%e A308838 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.
%o A308838 (Sage)
%o A308838 def msos_search(F, single=False):
%o A308838     squares = {x^2 for x in F}
%o A308838     MSOS = []
%o A308838     E = 0
%o A308838     for A, I in Subsets(squares, 2):
%o A308838         if A + I != 2*E: continue
%o A308838         C, G = 1, -1
%o A308838         B = 3*E - A - C
%o A308838         D = 3*E - A - G
%o A308838         F = 3*E - C - I
%o A308838         H = 3*E - G - I
%o A308838         if len(squares & {B,D,F,H}) < 4: continue
%o A308838         if len({A,B,C,D,E,F,G,H,I}) < 9: continue
%o A308838         if single: return [A,B,C,D,E,F,G,H,I]
%o A308838         MSOS.append([A,B,C,D,E,F,G,H,I])
%o A308838     E = 1
%o A308838     sequences = []
%o A308838     for A, I in Subsets(squares, 2):
%o A308838         if A + I != 2*E: continue
%o A308838         for C, G in sequences:
%o A308838             B = 3*E - A - C
%o A308838             D = 3*E - A - G
%o A308838             F = 3*E - C - I
%o A308838             H = 3*E - G - I
%o A308838             if len(squares & {B,D,F,H}) < 4: continue
%o A308838             if len({A,B,C,D,E,F,G,H,I}) < 9: continue
%o A308838             if single: return [A,B,C,D,E,F,G,H,I]
%o A308838             MSOS.append([A,B,C,D,E,F,G,H,I])
%o A308838         sequences.append((A,I))
%o A308838     return MSOS
%o A308838 for q in range(3, 500, 2):
%o A308838     if len(factor(q)) > 1: continue
%o A308838     print(q, msos_search(GF(q), single=True))
%Y A308838 Cf. A309810, A348263, A364264.
%K A308838 nonn,hard,more
%O A308838 1,1
%A A308838 _Onno M. Cain_, Jun 27 2019