First we verify that $2592$ does indeed satisfy the given property. {{begin-eqn}} {{eqn | l = 2592       | r = 2^5 \times 3^4       | c = Prime Decomposition of $2592$ }} {{eqn | r = 2^5 \times \paren {3^2}^2       | c =  }} {{eqn | r = 2^5 \times 9^2       | c =  }} {{end-eqn}} {{qed|lemma}} It remains to be shown that this is the only such number. Because $\sqbrk {abcd} = a^b \times c^d$, neither $a^b$ nor $c^d$ can have more than $4$ digits. Hence, for each digit, the highest power is: {{begin-eqn}} {{eqn | l = 1^9       | r = 1 }} {{eqn | l = 2^9       | r = 512 }} {{eqn | l = 3^8       | r = 6561 }} {{eqn | l = 4^6       | r = 4096 }} {{eqn | l = 5^5       | r = 3125 }} {{eqn | l = 6^5       | r = 7776 }} {{eqn | l = 7^4       | r = 2401 }} {{eqn | l = 8^4       | r = 4096 }} {{eqn | l = 9^4       | r = 6561 }} {{end-eqn}} Neither $a$ or $c$ can be zero, or that would make $\sqbrk {abcd} = 0$. Suppose $a = 1$ or $b = 0$. Then: :$\sqbrk {abcd} = c^d$ Apart from the above powers which have $4$ digits, we also have: {{begin-eqn}} {{eqn | l = 3^7       | r = 2187 }} {{eqn | l = 4^5       | r = 1024 }} {{eqn | l = 6^4       | r = 1296 }} {{end-eqn}} Hence, by inspection, it is seen that none of these fit the pattern $\sqbrk {1bcd}$ or $\sqbrk {a0cd}$. Similarly, suppose $c = 1$ or $d = 0$. Then: :$\sqbrk {abcd} = a^b$ Again, by inspection, it is seen that none of the above $4$-digit powers fit the pattern $\sqbrk {ab1d}$ or $\sqbrk {abc0}$. Suppose either $a^b$ or $c^d$ has $4$ digits. Then the other is less than $10$, giving: {{begin-eqn}} {{eqn | l = n^1       | r = n }} {{eqn | l = 2^2       | r = 4 }} {{eqn | l = 2^3       | r = 8 }} {{eqn | l = 3^2       | r = 9 }} {{end-eqn}} We try multiplying these by all the above $4$-digit powers such that their product is $4$ digits (there are not many). First note that $1^1$ can be ruled out as none of these $4$-digit powers either begins or ends with $11$. {{begin-eqn}} {{eqn | l = 3^7 \times 2^1       | r = 2187 \times 2 }} {{eqn | r = 4374 }} {{eqn | l = 3^7 \times 3^1       | r = 2187 \times 3 }} {{eqn | r = 6561 }} {{eqn | l = 3^7 \times 4^1       | r = 3^7 \times 2^2 }} {{eqn | r = 2187 \times 4 }} {{eqn | r = 8748 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 4^5 \times 2^1       | r = 1024 \times 2 }} {{eqn | r = 2048 }} {{eqn | l = 4^5 \times 3^1       | r = 1024 \times 3 }} {{eqn | r = 3072 }} {{eqn | l = 4^5 \times 4^1       | r = 4^5 \times 2^2 }} {{eqn | r = 1024 \times 4 }} {{eqn | r = 4096 }} {{eqn | l = 4^5 \times 5^1       | r = 1024 \times 5 }} {{eqn | r = 5120 }} {{eqn | l = 4^5 \times 6^1       | r = 1024 \times 6 }} {{eqn | r = 6144       | c = a near miss: $6^1 \times 4^5 = 6144$ }} {{eqn | l = 4^5 \times 7^1       | r = 1024 \times 7 }} {{eqn | r = 7168 }} {{eqn | l = 4^5 \times 8^1       | r = 4^5 \times 2^3 }} {{eqn | r = 1024 \times 8 }} {{eqn | r = 8192 }} {{eqn | l = 4^5 \times 9^1       | r = 4^5 \times 3^2 }} {{eqn | r = 1024 \times 9 }} {{eqn | r = 9216 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 4^6 \times 2^1       | r = 4096 \times 2 }} {{eqn | r = 8192 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 5^5 \times 2^1       | r = 3125 \times 2 }} {{eqn | r = 6250 }} {{eqn | l = 5^5 \times 3^1       | r = 3125 \times 3 }} {{eqn | r = 9375 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6^4 \times 2^1       | r = 1296 \times 2 }} {{eqn | r = 2592 }} {{eqn | l = 6^4 \times 3^1       | r = 1296 \times 3 }} {{eqn | r = 3888 }} {{eqn | l = 6^4 \times 4^1       | r = 6^4 \times 2^2 }} {{eqn | r = 1296 \times 4 }} {{eqn | r = 5184 }} {{eqn | l = 6^4 \times 5^1       | r = 1296 \times 5 }} {{eqn | r = 6480 }} {{eqn | l = 6^4 \times 6^1       | r = 1296 \times 6 }} {{eqn | r = 7776 }} {{eqn | l = 6^4 \times 7^1       | r = 1296 \times 7 }} {{eqn | r = 9072 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 7^4 \times 2^1       | r = 2401 \times 2 }} {{eqn | r = 4802 }} {{eqn | l = 7^4 \times 3^1       | r = 2401 \times 3 }} {{eqn | r = 7203 }} {{eqn | l = 7^4 \times 4^1       | r = 7^4 \times 2^2 }} {{eqn | r = 2401 \times 4 }} {{eqn | r = 9604 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 8^4 \times 2^1       | r = 4096 \times 2 }} {{eqn | r = 8192 }} {{end-eqn}} So none of the above $4$-digit powers multiplied by any of these single-digit powers satisfies the condition. We have that $\sqbrk {abcd}$ cannot end in $0$, as that can also be ruled out by inspection of the $4$-digit powers. Similarly, because $a \ne 1$, $\sqbrk {abcd} > 2000$. Also, it cannot be the case that both $a^b$ and $c^d$ are smaller than $44$, as $44^2 < 2000$. {{finish}} {{Namedfor|Henry Ernest Dudeney|cat = Dudeney}}