If you just use powers of 2, the minimal square with distinct entries has multiplicative constant 1073741824 (2^30). 55440 is much smaller. Is it the smallest?
For multiplicative magic squares it seems reasonable to allow negative numbers and ask for the minimum absolute value. So you could replace 11 with -1 in your example, giving magic product -5040 . Magic products with other -1 patterns might allow further reductions. If you look at the projections of primes (or unit -1) and consider patterns needed to ensure all entries are unique, information theory gives some lower bounds. What do they say? Are there cases where the information theoretic bound can't be achieved? I conjecture yes. Is this question solved for the simpler problem of dropping the magic constraint on diagonals? That question sounds interesting too. Best, - Scott
--Ed Pegg Jr
--- "Torgerson, Mark D" <mdtorge@sandia.gov> wrote:
Replacement goes a long way. You may take all the 3s in the square below and turn them into 29s. Or some other square having large primes may be replaced with a square with smaller corresponding primes. Even primes within a particular square may be switched. 55440=2^4*3^2*5*7*11 so you can make a corresponding square with 2*3^2*5*7*11^4. No matter the starting square, this reduction leads to some sort of minimal representation, where the factorization of the product gives 2 with the largest exponent, 3 the next and so on.