An additive-multiplicative magic square is a square array of distinct positive integers such that
- every row, every column, and each of two diagonals all give the same sum $S$,
- every row, every column, and each of two diagonals all give the same product $P$.
My question is about what happens if the requirement "positive integers" is weakened to "nonzero complex numbers".
Is there a $5\times 5$ or $6\times 6$ "generalized additive-multiplicative magic square" whose entries are distinct nonzero complex numbers? What approaches or simplifications can be used to find solutions?
I unsuccessfully tried to use a CAS to attack the $5\times 5$ case by constructing a system of 24 equations with 27 unknowns ($12=5+5+2$ equations of the form "the sum of five entries is $S$" and 12 more equations of the form "the product of five entries is $P$"). It is not hard to eliminate entries in the last row and the last column through the equations giving sums of entries in a row/column, leaving only 18 unknowns ($S$, $P$ and the remaining 16 entries). If the central $3\times 3$ square is known, I think one can recover $S$, $P^3$, the sum of corners and the cubed product of corners, $$ \begin{align} 3\cdot S &= 3\cdot a_{33}+a_{23}+a_{32}+a_{34}+a_{43}+2\cdot\left(a_{22}+a_{24}+a_{42}+a_{44}\right) \\ 3\cdot\left(a_{11}+a_{15}+a_{51}+a_{55}\right) &= a_{22}+a_{24}+a_{42}+a_{44}+2\cdot\left(a_{23}+a_{32}+a_{34}+a_{43}\right) \\ P^{3} &= a_{33}^{3}\cdot a_{23}\cdot a_{32}\cdot a_{34}\cdot a_{43}\cdot\left(a_{22}\cdot a_{24}\cdot a_{42}\cdot a_{44}\right)^{2} \\ \left(a_{11}\cdot a_{15}\cdot a_{51}\cdot a_{55}\right)^{3} &= a_{22}\cdot a_{24}\cdot a_{42}\cdot a_{44}\cdot\left(a_{23}\cdot a_{32}\cdot a_{34}\cdot a_{43}\right)^{2} \end{align} $$ $$ \begin{array}{|ccccc|} \hline a_{11} & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\\hline \end{array} $$
However, so far I was unable to either get a solution with distinct entries from the resulting system or prove impossibility.
According to (1) (2), it is currently unknown whether there are $5\times 5$ or $6\times 6$ additive-multiplicative magic squares (whose entries are distinct positive integers). It is known that $3\times 3$ and $4\times 4$ additive-multiplicative magic squares are impossible. In August 2016 Sébastien Miquel discovered the following $7\times 7$ additive-multiplicative magic square with $S = 465$ and $P = 150{,}885{,}504{,}000$:
$$ \begin{array}{|ccccccc|} \hline 126 & 66 & 50 & 90 & 48 & 1 & 84\\ 20 & 70 & 16 & 54 & 189 & 110 & 6\\ 100 & 2 & 22 & 98 & 36 & 72 & 135\\ 96 & 60 & 81 & 4 & 10 & 49 & 165\\ 3 & 63 & 30 & 176 & 120 & 45 & 28\\ 99 & 180 & 14 & 25 & 7 & 108 & 32\\ 21 & 24 & 252 & 18 & 55 & 80 & 15 \\\hline \end{array} $$
