Imagine we start with a 3 by 3 square using the numbers 1,...,9. We then multiply the numbers across each row and note their products as the set {x1, x2, x3}. We then multiply the numbers across each column and note their products as the set {x4, x5, x6}. My question is whether or not is possible to create a magic square such that {x1, x2, x3}={x4, x5, x6}, in other words where the set of the values obtained by multiplying across every row equals the set of values obtained by multiplying across every column.
My intuition tells me that the unique prime factors should lie on either main diagonal (by unique prime factors, in the 3 by 3 case, I mean 5 and 7, since they are prime and are not a factor of any of the other numbers 1,...9). I also think it's a good idea to consider how many prime factors we have. In the 3 by 3 case, we have seven 2's, four 3's, one 5, and one 7 available.
So far, I believe it is impossible to create such a 3 by 3 magic square. Although I'm not sure how I could prove this impossibility. Any thoughts or suggestions on how I could prove this impossible for the 3 by 3 case, or if there does happen to be such a configuration, would be greatly appreciated!
Thank you!
