On 9/21/05, Michael Kleber <michael.kleber@gmail.com> wrote:
Just blathering off the top of my head:
Ed pegg wrote:
Are multiplicative magic squares well studied? For example, using dots merely as spacers, the following has a multiplicative constant of 55440.
231 .40 ..3 ..2 ..6 ..1 132 .70 ..4 .21 .20 .33 .10 .66 ..7 .12
For any prime p, this must still be multiplicatively magic if you replace each entry with the largest power of p dividing it. And that must be additively magic if you take its log_p. For instance, for p=2, the above becomes
0 3 0 1 1 0 2 1 2 0 2 0 1 1 0 2
with additive constant 4, the power of 2 in 55440.
Of course, this projection destroys distinctness of entries. For that you just need to find another one like the above such that the ordered pairs of corresponding entries (a,b) are all distinct, and then make a multiplicative one out of 2^a 3^b.
Well, 2^a q^b anyway, where q is the prime that makes them all distinct.
--Michael Kleber
-- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike