Dudeney's Property of 2592

From ProofWiki
Jump to navigation Jump to search

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 $44^2 < 2000$.



This needs considerable tedious hard slog to complete it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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 term Dudeney's Property of 2592 was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ as a convenient shorthand for what would otherwise be tediously unwieldy.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Dudeney%27s_Property_of_2592&oldid=475819"