$3 \times 3$ matrix with determinant a large power of $2$

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}

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

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

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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 $2^{21}$.

I noticed that we can infer the least prime p such that determinant is $2^n$ by using the relation determinant <$2p^3.$
The least prime will satisfy the inequality $2^n<2p^3<2^{(n+1)}.$

        expected range   actual prime  
2^20:    80-101            101
2^21:    101-128           127
2^22:    128-161           157
2^30:    812-1024          853
2^31:    1024-1290          ?

$$\pmatrix{a & b & c\cr d & e & f\cr g & h & i}$$

Above determinant is given below.

$$a(ei - fh) - b(di - fg) + c(dh - eg)$$

Let's assume $3 \times 3$ matrix with entries are distinct positive integer $<100.$
Maximum value of determinant $2 \times 2$ matrix<$99^2<2.4\times2^{12}.$
$3 \times 3$ matrix has the following properties.
If $(ei - fh)$ and $(dh - eg)$ are positive, $(di - fg)$ will always be positive. I checked that by brute force.
Hence we get

\begin{align} a(ei - fh) - b(di - fg) + c(dh - eg)&<(a+c)\times 2.4\times2^{12}\\ &<200\times2.4\times2^{12}\\ &<2^{21}\\ \end{align}

Hence maximum $2^n$ of determinant $3 \times 3$ matrix=$2^{20}.$

Result of brute force search.
$3 \times 3$ matrix with entries are distinct primes $<100.$

Case: Not all entries are prime.
Determinant is $2^{20}$.

$$\pmatrix{71 & 3 & 89\cr 83 & 79 & 13\cr 11 & 97 & 93}$$ $$\pmatrix{73 & 5 & 83\cr 89 & 79 & 11\cr 7 & 91 & 93}$$ $$\pmatrix{83 & 13 & 79\cr 89 & 71 & 5\cr 3 & 93 & 95}$$

Case: All entries are prime.

It seems that there is no solution for determinant is $2^n$ with $n>19.$
There are many matrices with determinant is $2^{19}.$
For example,
$$\pmatrix{3 & 5 & 79\cr 83 & 13 & 23\cr 29 & 89 & 71}$$ $$\pmatrix{3 & 11 & 79\cr 83 & 7 & 59\cr 39 & 89 & 53}$$ $$\pmatrix{39 & 3 & 59\cr 89 & 61 & 5\cr 7 & 79 & 71}$$