Let $n\ge 1$ be a natural number. Arrange the first $n^2$ primes in a $n\times n$-matrix, such that the determinant becomes as large as possible.
- What is the largest possible determinant and which matrix has it ?
For $n=2$, we have $\pmatrix{7&3\\2&5}$ with determinant $29$, which is optimal.
For $n=3$, we have $\pmatrix{23&7&3\\5&19&13\\11&2&17}$ with determinant $6640$, which is optimal.
For $n=4$, the best value I found is $\pmatrix{37&7&47&23\\17&19&11&53\\41&31&2&5\\3&43&29&13}$ with deteminant $4673460$.
This is A180128 in the OEIS. You can do better for $n=4:$ $$ \pmatrix{53&11&23&13\\ 17&47&29&3\\ 7&5&43&37\\ 19&31&2&41}=4868296>4673460. $$
The fifth term is $$ \pmatrix{89&41&23&2&53\\ 31&97&29&47&11\\ 59&13&79&61&7\\ 37&19&5&83&67\\ 3&43&71&17}=735725998504 $$ and the sixth term is $$ \pmatrix{137&73&7&89&83&13\\ 79&139&67&19&3&97\\ 101&5&149&61&37&53\\ 2&109&103&71&113&11\\ 59&29&41&17&131&127\\ 23&47&43&151&31&107}=11305600374272. $$
The seventh term is believed to be 35954639671827332, and the eighth term is between 154665569137423060000 and 154715716383037989022.
The terms were found by Hugo Pfoertner using tabu search. The upper bound is due to Gasper, Pfoertner, & Sigg (2009).