A348263 Orders of Parker fields.

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

Author

Thomas Scheuerle, Oct 09 2021

Keywords

• nonn
• more

Comments

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

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A308838, A309810, A364264.

Extensions

Missing even terms added by Yevhenii Diomidov, Jan 19 2022