Re: [math-fun] 6x6 pandiag. multiplic. squares ARE NOT impossible!
Christian writes: << Dan, you are right, there is a great relationship between additive and multiplicative squares (and cubes), but not always a direct relationship.
I may have been less than clear. Considering only conditions on rows & columns, all I was saying is that, suppose p_k is the largest prime divisor of any entry in a MuMaSq: Then if each entry p_1^e_1 * . . . * p_k^e_k (all e_j >= 0) is replaced by its tuple of exponents (e_1,...,e_k), the result is an AdMaSq -- not of mere integers, but of k-tuples of integers. Or if you prefer, k ordinary magic squares (possibly with repeated entries), one for each j in {1,...,k}, each created by replacing each entry in the MuMaSq by its jth exponent, e_j. And conversely, though extra care must be taken to assure that all final entries be distinct. --Dan
Dan, I better understand what you meant about the relationship between additive and multiplicative squares. But, as you wrote, "ordinary magic squares possibly with repeated entries", and "extra care must be taken to assure that all final entries be distinct" in multiplicative squares. Using your split, and as remarked by Michael, my 6x6 pandiagonal magic square is coming from two matrixes, one for 2^: 5 0 4 5 0 4 1 6 2 1 6 2 5 0 4 5 0 4 1 6 2 1 6 2 5 0 4 5 0 4 1 6 2 1 6 2 and its transpose for 3^: 5 1 5 1 5 1 0 6 0 6 0 6 4 2 4 2 4 2 5 1 5 1 5 1 0 6 0 6 0 6 4 2 4 2 4 2 Difficult to call them magic squares because they use (a lot of) repeated entries. Of course, it is trivial to construct a multiplicative magic square from any normal additive magic square, replacing each entry n by 2^n. But very difficult to see a general construction method of the best possible multiplicative magic squares: no known direct correspondence between additive squares <-> multiplicative squares with THE smallest possible integers and product. And find these best possible multiplicative squares (or cubes) are the interesting question. Christian.
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Christian Boyer -
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