Sequence of Dudeney Numbers

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Theorem

The only Dudeney numbers are:

$0, 1, 8, 17, 18, 26, 27$

two of which are themselves cubes, and one of which is prime.

This sequence is A046459 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

We have trivially that:

\(\ds 0^3\) \(=\) \(\ds 0\)
\(\ds 1^3\) \(=\) \(\ds 1\)


Then:

\(\ds 8^3\) \(=\) \(\ds 512\)
\(\ds 8\) \(=\) \(\ds 5 + 1 + 2\)


\(\ds 17^3\) \(=\) \(\ds 4913\)
\(\ds 17\) \(=\) \(\ds 4 + 9 + 1 + 3\)


\(\ds 18^3\) \(=\) \(\ds 5832\)
\(\ds 18\) \(=\) \(\ds 5 + 8 + 3 + 2\)


\(\ds 26^3\) \(=\) \(\ds 17576\)
\(\ds 26\) \(=\) \(\ds 1 + 7 + 5 + 7 + 6\)


\(\ds 27^3\) \(=\) \(\ds 19683\)
\(\ds 27\) \(=\) \(\ds 1 + 9 + 6 + 8 + 3\)


A quick empirical test shows that when $n = 46$, it is already too large to be the sum of the digits of its cube.


For $46 < n \le 54$, $n^3 \le 54^3 < 200 \, 000$.

Hence the sum of the digits of $n^3$ is less than:

$1 + 5 \times 9 = 46 < n$


For $54 < n < 100$, $n^3 < 10^6$.

Hence the sum of the digits of $n^3$ is less than:

$6 \times 9 = 54 < n$


For $n \ge 100$, let $n$ be a $d$-digit number, where $d \ge 3$.

Then $10^{d - 1} \le n < 10^d$ and $n^3 < 10^{3 d}$.

Hence the sum of the digits of $n^3$ is less than:

\(\ds 3 d \times 9\) \(=\) \(\ds 27 d\)
\(\ds \) \(\le\) \(\ds 27 d + 63 d - 189\)
\(\ds \) \(<\) \(\ds 90 d - 170\)
\(\ds \) \(=\) \(\ds 10 \paren {1 + 9 \paren {d - 2} }\)
\(\ds \) \(\le\) \(\ds 10 \times \paren {1 + 9}^{d - 2}\) Bernoulli's Inequality
\(\ds \) \(\le\) \(\ds n\)

so no numbers greater than $46$ can have this property.

$\blacksquare$


Also reported as

Some sources (either deliberately or by oversight) do not include $0$ in this list.


Also see


Historical Note

The earliest known appearance of this result is from Claude Séraphin Moret-Blanc in $1879$, although exactly where this was published is still to be identified.

Henry Ernest Dudeney subsequently published it in one of his own collections.

As a result, a number which equals the sum of the digits of its cube is now called a Dudeney number.

It continues to crop up occasionally in publications devoted to recreational mathematics.


Sources