Let $p$ be a large prime number. I want a $k\times k$ matrix with determinant $p$ and bounded integer elements (say, from -100 to 100). For which minimal $k$ such a matrix does always exist? We can not hope for anything better than $k=O(\log p/\log\log p)$, which corresponds to $p\sim k^{Ck}$, and we may always achieve $k=O(\log p)$ if I understood correctly the answers to [this][1] question (determinant of (0,1) $k\times k$ matrices achieve all values between 0 and something exponential in $k$.)


  [1]: http://mathoverflow.net/questions/18547/number-of-unique-determinants-for-an-nxn-0-1-matrix