Please, Christian, tell us how you created that list. --Michael Kleber On 9/26/05, Christian Boyer <cboyer@club-internet.fr> wrote:
Sorry Michael, I have not completely read one of your previous messages, you mentioned already the same square:
(p,q,pq; pp,qq,r) [4,4,1] 6480
-----Message d'origine----- De: Christian Boyer [mailto:cboyer@club-internet.fr] Envoyé: lundi 26 septembre 2005 16:14 À: 'math-fun' Objet: RE: [math-fun] Multiplicative Magic Squares
From: "Michael Kleber" <michael.kleber@gmail.com>
So abcdABCD makes a pretty small fraction of all multiplicative magic squares, as the magic product (and the exponents on each prime in its factorization) get large.
Here is an interesting family of 4x4 multiplicative squares, using only three variables a, b, c:
1 abbb bc aaa ac aab ab bb aaab c abb b bbb a aa abc
Magic product = a^4 * b^4 * c, for the 4 rows, 4 columns, 2 diagonals and 2 broken diagonals. And also for other sets of four cells (the corners for example, the 4 numbers of each quarter, ...).
Apply a=2 and b=3, and you will get one of the various possible examples with magic product 6480. 6480 is the second smallest product after 5040, as listed in my previous email.
Christian.
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