Christian Boyer wrote:
Here is one of the numerous possible examples with 302400: 12 35 1 40 18 36 2 24 7 25 14 45 15 4 8 5 16 42 30 3 10 6 20 9 28
Very nice! Thank you, Christian. (Just to be clear, your "numerous examples" include ones with a different set of numbers, and not the same ones in another arrangement, right?) We can look at your square through the Hilbert basis lens -- that is, (1) decompose it as a pointwise product of squares all of whose entries are powers of p, for each of p=2,3,5,7, and then take the pointwise log_p of each, to get four additive magic squares with repeat numbers, and then (2) ask whether the resulting additive magic squares are "irreducible" or are smaller magic squares on top of each other. Of course for p=7 you just get a single magic permutation matrix, so there is nothing to do. (Turns out that it is, in fact, the "satin" matrix from which I built my example.) For p=5 and for p=3 the matrices have magic sums 2 and 3 resp, and I just checked by hand: both of these are irreducible. This is in contrast to the 4x4 case, where all the matrices that arose from the 5040 example could be decomposed into sum-1 permutation matrices. But that's not really surprising: the Ahmed/De Loera/Hemmecke paper showed that in 4x4, all irreducibles are sum 1 or 2, with 8 and 12 of each, respectively; while in 5x5 there are 4828 irreducibles, only 20 of which have sum 1. Indeed, if the 5x5 had been built out of degree-1 generations, *that* would have demanded some explanation! --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.