David Wilson wrote:
A paper on the minimal 4x4 magic product would be a nice addition to the JIS.
It seems clear to me that the logic we've been using will be enough to completely describe (with only finitely much work) the sequence "All possible magic products of 4x4 multiplicative magic squares". Surely that's the desired paper. As we've made clear, the property only depends on the exponents in the prime factorization of the potential magic product. Moreover any multiple of a magic product is again one, since we can multiply any magic square pointwise by any of eight rotations or reflections of n 1 1 1 1 1 n 1 1 1 1 n 1 n 1 1 Oh, wait, it's not quite that easy -- it's not immediately clear that you can always find one of these that still has no duplicate entries. Hmm. Ignoring that for the moment, the question boils down to what the minimal exponent sets are that allow magic squares. The 5040 square shows that signature [4,2,1,1] works, and the same construction will give lots of others, so let's make it clear: Prop 1. For any 6-tuple (a1,a2,a3;b1,b2,b3), there is a multiplicative magic square whose sixteen entries are the pairwise products ai*bj, 0<=i,j<=3, where a0=b0=1. Its magic product is a1*a2*a3*b1*b2*b3. (Of course, one still needs to check that these entires are all distinct.) The 5040 example is concisely expressed by the 6-tuple (p,q,pq; pp,r,s), where each letter represents a distinct prime. We can construct a table of signatures attainable this way: 6-tuple prod. signature (minimum magic product) (p,q,r; s,t,u) [1,1,1,1,1,1] 30030 (p,pp,q; r,s,t) [3,1,1,1,1] 9240 (p,pp,q; r,rr,s) [3,3,1,1] 7560 (p,q,pq; r,s,t) [2,2,1,1,1] 13860 (p,q,pq; r,s,rs) [2,2,2,2] 44100 (p,q,pq; pp,r,s) [4,2,1,1] 5040 (p,q,pq; pp,qq,r) [4,4,1] 6480 (p,q,r; pp,qq,rr) [3,3,3] 27000 ...etc... (p,pp,ppp; p^4,p^8,p^12) [30] 1073741824 There's no intelligent order here; I'm just doodling by hand. But a computer could easily take any tentative product signature and try to factor it as a 6-tuple with 16 distinct pairwise products, and fill this table out completely This gives us lots of magic squares that do exist. Contrarywise, to find ones that don't exist, we implement David's "needs 16 factors with plausible quotients" rule, and start ruling out prime signatures that are impossible. This criterion can be extended a bit, too: eight of the 16 entires in the magic square participate in a row, column, and diagonal, so the quotient has to be writable as a product of three terms in three ways, with all nine terms distinct. That will rule out lots of prime signatures. Between those two techniques, there should only be finitely many signatures that are possible but not yet constructed, and all small enough that an exhaustive search will finish the problem off. So that's fuzzy, but seems a plausible plan, if there are folks with the time to play with it. (Sadly I don't count these days.) --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.