Sequence of Dudeney Numbers
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
- Definition:Armstrong Number, with which the numbers in this entry appear frequently to be conflated
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
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $69$. -- Root Extraction
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $120$. Root Extraction
- 1979: S.P. Mohanty and Hemant Kumar: Powers of Sums of Digits (Math. Mag. Vol. 52: pp. 310 – 312) www.jstor.org/stable/2689785
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables: $27$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $17$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $18$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $26$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $27$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4913$
- 1992: Joe Roberts: Lure of the Integers: $27$
- 1993: Monte James Zerger: The 'Number of Mathematics' (Journal of Recreational Mathematics Vol. 25, no. 4: pp. 247 – 251)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $18$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $26$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4913$
- Weisstein, Eric W. "Cubic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicNumber.html