Dudeney's Property of 2592

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Theorem

$2592 = 2^5 \times 9^2$


It is the only number $n$ that has the property that:

$n = \sqbrk {abcd} = a^b \times c^d$

where $\sqbrk {abcd}$ denotes the decimal representation of $n$.


Proof

First we verify that $2592$ does indeed satisfy the given property.

\(\ds 2592\) \(=\) \(\ds 2^5 \times 3^4\) Prime Decomposition of $2592$
\(\ds \) \(=\) \(\ds 2^5 \times \paren {3^2}^2\)
\(\ds \) \(=\) \(\ds 2^5 \times 9^2\)

$\Box$


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:

\(\ds 1^9\) \(=\) \(\ds 1\)
\(\ds 2^9\) \(=\) \(\ds 512\)
\(\ds 3^8\) \(=\) \(\ds 6561\)
\(\ds 4^6\) \(=\) \(\ds 4096\)
\(\ds 5^5\) \(=\) \(\ds 3125\)
\(\ds 6^5\) \(=\) \(\ds 7776\)
\(\ds 7^4\) \(=\) \(\ds 2401\)
\(\ds 8^4\) \(=\) \(\ds 4096\)
\(\ds 9^4\) \(=\) \(\ds 6561\)

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:

\(\ds 3^7\) \(=\) \(\ds 2187\)
\(\ds 4^5\) \(=\) \(\ds 1024\)
\(\ds 6^4\) \(=\) \(\ds 1296\)

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:

\(\ds n^1\) \(=\) \(\ds n\)
\(\ds 2^2\) \(=\) \(\ds 4\)
\(\ds 2^3\) \(=\) \(\ds 8\)
\(\ds 3^2\) \(=\) \(\ds 9\)


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 $11$.


\(\ds 3^7 \times 2^1\) \(=\) \(\ds 2187 \times 2\)
\(\ds \) \(=\) \(\ds 4374\)
\(\ds 3^7 \times 3^1\) \(=\) \(\ds 2187 \times 3\)
\(\ds \) \(=\) \(\ds 6561\)
\(\ds 3^7 \times 4^1\) \(=\) \(\ds 3^7 \times 2^2\)
\(\ds \) \(=\) \(\ds 2187 \times 4\)
\(\ds \) \(=\) \(\ds 8748\)


\(\ds 4^5 \times 2^1\) \(=\) \(\ds 1024 \times 2\)
\(\ds \) \(=\) \(\ds 2048\)
\(\ds 4^5 \times 3^1\) \(=\) \(\ds 1024 \times 3\)
\(\ds \) \(=\) \(\ds 3072\)
\(\ds 4^5 \times 4^1\) \(=\) \(\ds 4^5 \times 2^2\)
\(\ds \) \(=\) \(\ds 1024 \times 4\)
\(\ds \) \(=\) \(\ds 4096\)
\(\ds 4^5 \times 5^1\) \(=\) \(\ds 1024 \times 5\)
\(\ds \) \(=\) \(\ds 5120\)
\(\ds 4^5 \times 6^1\) \(=\) \(\ds 1024 \times 6\)
\(\ds \) \(=\) \(\ds 6144\) a near miss: $6^1 \times 4^5 = 6144$
\(\ds 4^5 \times 7^1\) \(=\) \(\ds 1024 \times 7\)
\(\ds \) \(=\) \(\ds 7168\)
\(\ds 4^5 \times 8^1\) \(=\) \(\ds 4^5 \times 2^3\)
\(\ds \) \(=\) \(\ds 1024 \times 8\)
\(\ds \) \(=\) \(\ds 8192\)
\(\ds 4^5 \times 9^1\) \(=\) \(\ds 4^5 \times 3^2\)
\(\ds \) \(=\) \(\ds 1024 \times 9\)
\(\ds \) \(=\) \(\ds 9216\)


\(\ds 4^6 \times 2^1\) \(=\) \(\ds 4096 \times 2\)
\(\ds \) \(=\) \(\ds 8192\)


\(\ds 5^5 \times 2^1\) \(=\) \(\ds 3125 \times 2\)
\(\ds \) \(=\) \(\ds 6250\)
\(\ds 5^5 \times 3^1\) \(=\) \(\ds 3125 \times 3\)
\(\ds \) \(=\) \(\ds 9375\)


\(\ds 6^4 \times 2^1\) \(=\) \(\ds 1296 \times 2\)
\(\ds \) \(=\) \(\ds 2592\)
\(\ds 6^4 \times 3^1\) \(=\) \(\ds 1296 \times 3\)
\(\ds \) \(=\) \(\ds 3888\)
\(\ds 6^4 \times 4^1\) \(=\) \(\ds 6^4 \times 2^2\)
\(\ds \) \(=\) \(\ds 1296 \times 4\)
\(\ds \) \(=\) \(\ds 5184\)
\(\ds 6^4 \times 5^1\) \(=\) \(\ds 1296 \times 5\)
\(\ds \) \(=\) \(\ds 6480\)
\(\ds 6^4 \times 6^1\) \(=\) \(\ds 1296 \times 6\)
\(\ds \) \(=\) \(\ds 7776\)
\(\ds 6^4 \times 7^1\) \(=\) \(\ds 1296 \times 7\)
\(\ds \) \(=\) \(\ds 9072\)


\(\ds 7^4 \times 2^1\) \(=\) \(\ds 2401 \times 2\)
\(\ds \) \(=\) \(\ds 4802\)
\(\ds 7^4 \times 3^1\) \(=\) \(\ds 2401 \times 3\)
\(\ds \) \(=\) \(\ds 7203\)
\(\ds 7^4 \times 4^1\) \(=\) \(\ds 7^4 \times 2^2\)
\(\ds \) \(=\) \(\ds 2401 \times 4\)
\(\ds \) \(=\) \(\ds 9604\)


\(\ds 8^4 \times 2^1\) \(=\) \(\ds 4096 \times 2\)
\(\ds \) \(=\) \(\ds 8192\)


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 $44^2 < 2000$.





Source of Name

This entry was named for Henry Ernest Dudeney.


Historical Note

This result is generally attributed to Henry Ernest Dudeney, who stated it in his $1917$ book Amusements in Mathematics.

It continues to crop up occasionally in puzzle pages of journals.

The name Dudeney's Property of 2592 has hence been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$, as a convenient shorthand for what would otherwise be tediously unwieldy.


Sources