cp's OEIS Frontend

Named after Matt Parker, who attempted (and failed) to create a 3 X 3 magic square of squares (still an open problem). The sum of entries in the rows, columns and one diagonal is 3051, but in the other diagonal the sum is 4107. Moreover, three entries are repeated (1^2, 29^2 and 41^2).

Cain (2019) cites this trivial semimagic square and calls a finite field a Parker field if no 3 X 3 magic square of squares can be constructed using 9 distinct squared elements.

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 ring of order 31.

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