TITLE: $3 \times 3$ matrix with determinant a large power of $2$
QUESTION [36 upvotes]: Here's a little curiosity I found.  The following $3 \times 3$ matrix has entries that are distinct primes $< 100$ and its determinant is $2^{19}$.
$$ \pmatrix{71 & 31 & 97\cr 61 & 67 & 23\cr 7 & 83 & 73}$$
Does anyone know of another such matrix whose determinant is nonzero and divisible by a larger power of $2$?  If not, what is the least $p$ such that there is a $3 \times 3$ matrix with entries distinct primes $\le p$ and determinant nonzero and divisible by a larger power of $2$?  What if we drop the requirement of the entries being primes (and allow distinct positive integers $\le p$)?

REPLY [20 votes]: To get determinant $2^{20}$, we require to include one additional prime number: $101$, and one of such examples will be:
$$ \det \pmatrix{101 & 17 & 67 \cr 61 & 97 & 3\cr 7 & 83 & 89} = 2^{20}.$$
(updated)
Here is list of examples of matrices with distinct prime entries with determinants of the form $2^d$ (note that I'm not sure that those $p_{max}$ are absolutely correct: some of them probably could be improved a bit):
\begin{array}{|c|c|c|}
\hline
d & p_{max}& matrix & det \\
\hline
20 & 101 & \pmatrix{101 & 79 & 2 \cr 13 & 83 & 89 \cr 71 & 17 & 97} & 2^{20} \\
\hline
21 & 127 & \pmatrix{127 & 107 & 19 \cr 3 & 109 & 103 \cr 89 & 5 & 101} & 2^{21}\\
\hline
22 & 151 & \pmatrix{151 & 139 & 13 \cr 3 & 127 & 137 \cr 103 & 19 & 149} & 2^{22}\\
\hline
23 & 181 & \pmatrix{181 & 167 & 3 \cr 11 & 157 & 179 \cr 163 & 13 & 151} & 2^{23}\\
\hline
24 & 229 & \pmatrix{229 & 193 & 13 \cr 7 & 191 & 227 \cr 181 & 3 & 223} & 2^{24}\\
\hline
25 & 277 & \pmatrix{277 & 241 & 3 \cr 7 & 269 & 271 \cr 257 & 29 & 263} & 2^{25} \\
\hline
26 & 349 & \pmatrix{349 & 317 & 3 \cr 5 & 337 & 331 \cr 313 & 13 & 311} & 2^{26} \\
\hline
27 & \underline{431} & \pmatrix{431 & 389 & 3 \cr 7 & 409 & 419 \cr 397 & 17 & 421} & 2^{27} \\
\hline
28 & 557 & \pmatrix{557 & 463 & 3 \cr 5 & 541 & 499 \cr 509 & 17 & 523} & 2^{28} \\
\hline
29 & 677 & \pmatrix{677 & 673 & 5 \cr 43 & 659 & 647 \cr 661 & 37 & 641} & 2^{29} \\
\hline
30 & 853 & \pmatrix{853 & 811 & 79 \cr 3 & 839 & 829 \cr 809 & 5 & 823} & 2^{30} \\
\hline
31 & 1063 & \pmatrix{1063 & 1051 & 3 \cr 13 & 1039 & 971 \cr 1049 & 43 & 1031} & 2^{31} \\
\hline
32 & 1321 & \pmatrix{1321 & 1289 & 17 \cr 31 & 1319 & 1279 \cr 1301 & 11 & 1291} & 2^{32} \\
\hline
\cdots \\
\end{array}

If drop the requirement of the entries being primes, and consider matrices with distinct entries (less than $100$), then there are examples with $2^{20}$:
$$ \det \pmatrix{99 & 81 & 10\cr 5 & 96 & 86\cr 87 & 26 & 82} = 2^{20},$$
$$ \det \pmatrix{96 & 94 & 6\cr 35 & 86 & 85\cr 91 & 8 & 87} = 2^{20};$$

odd (but not all prime) entries:
$$ \det \pmatrix{99 & 95 & 5\cr 23 & 93 & 89\cr 85 & 31 & 91} = 2^{20}.$$

It looks like the largest possible determinant of a matrix with distinct positive entries less than $100$ is $1742902\approx 2^{20.733}$ (or close to this number):
$$
\det\pmatrix{99 & 3 & 95\cr 94 & 98 & 1\cr 2 & 96 & 97} = 1742902,
$$
so there is no hope to find any $3\times 3$ such matrix with determinant $2^{21}$.