[math-fun] Multiplicative squares
Bonjour, As said yesterday, 302400 is the smallest possible product for 5x5 multiplicative squares. And 332640 is another new possible product smaller than our previous example with 362880. 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 coming from the family: aab cd 1 aaac abb aabb a aaab d cc ad bbc bc aa aaa c aaaa abd abc b ac ab aac bb aad magic product a^6 * b^3 * c^2 * d (example with a=2, b=3, c=5, d=7 for 302400, but of course the same family can be used for 423360, 475200, ...) Other numerous families are possible with the same magic product a^6 * b^3 * c^2 * d. Some words on the method used for nxn multiplicative squares. For each "good" P (means with several little factors), the program computes all the magic series of n distinct integers having the product P. For example with n=5: P=302400 has 3527 series, 332640 has 3661 series, 362880 has 3734 series. Then it analyzes combinations of n series looking if it is possible to get n series using n² distinct integers. If n series are found, they are the n rows: arrange them trying to get n columns and 2 diagonals. No great secret in the method... A specific page on multiplicative squares will be added in the next update of www.multimagie.com/indexengl.htm, planned beginning of October. Christian. ------------------------------------------ The 10 smallest magic products for 5x5 are: Prod. = 2^ * 3^ * 5^ * 7^ * 11^ *13^ 302400 6 3 2 1 0 0 << The SMALLEST 332640 5 3 1 1 1 0 362880 7 4 1 1 0 0 << Previously known 393120 5 3 1 1 0 1 << id 332640 403200 8 2 2 1 0 0 415800 3 3 2 1 1 0 423360 6 3 1 2 0 0 << id 302400 443520 7 2 1 1 1 0 475200 6 3 2 0 1 0 << id 302400 491400 3 3 2 1 0 1 << id 415800 ------------------------------------------
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.
From Michael Kleber Very nice! Thank you, Christian.
Thanks! It was a pleasure to work on the question.
(Just to be clear, your "numerous examples" include ones with a different set of numbers, and not the same ones in another arrangement, right?)
I meant numerous other examples using different set of 25 numbers, or using same set of 25 numbers but arranged through different series of 5 numbers. Another example with a wider set of numbers, from 1 to 100, but with the same magic product 302400: 3 28 1 100 36 25 16 27 4 7 24 9 35 2 20 14 5 40 18 6 12 15 8 21 10 Christian.
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Christian Boyer -
Michael Kleber