From Michael Kleber [ 10 24 4 42 9 ] [ 28 54 5 16 3 ] [ 8 2 21 36 30 ] [ 27 20 48 1 14 ] [ 6 7 18 15 32 ] diabolical magic product 9! = 362880 = 2^7.3^4.5.7
Once again, I feel like one ought to be able to argue from elementary principles that this must be the minimum possible, but I'm unable to do so (though I do think the Hilbert basis for the 4x4 case should now make that argument easy). So we can wait for Christian's algorithm to get fast enough to check everything smaller, or hope to use David Wilson's logic to limit the smaller possible signatures, etc.
Michael, very nice square! However I have a different feeling: I think (of course not sure) that it should be possible to find smaller 5x5 examples. But 362880 should be the smallest possible product of 5x5 pandiagonal magic square (or diabolic). Trying to find publications on multiplicative squares, I have the surprise to find... most of our squares... in the old book "Magic squares and cubes" by W.S. Andrews. Pages 283-294, there is a reprint of an excellent article -with construction methods- written by H.A. Sayles one century ago, date between 1905 and 1916. For example, we can find in the book: - 3x3 (216, 1000, 4096) - 4x4 (5040, 7560, pandiag 14400, ...) - 5x5 (pandiag 362880, pandiag 720720, ...) Good to see that different minds, even separated by one century, find same objects. 362880 seems to be currently the smallest known 5x5. Is it the smallest possible? I will keep you informed of my results (computing not yet started, free time is missing). Christian.