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A348263
Orders of Parker fields.
3
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
1,1
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
LINKS
Matt Parker, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).
EXAMPLE
The field GF(29), for example, is not Parker since:
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|9^2 |11^2|1^2 | mod 29 = 0
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|6^2 |0^2 |14^2| mod 29 = 0
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|12^2|16^2|8^2 | mod 29 = 0,
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with the same property for columns and main diagonals.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Thomas Scheuerle, Oct 09 2021
EXTENSIONS
Missing even terms added by Yevhenii Diomidov, Jan 19 2022
STATUS
approved