Prime Dudeney Number

Theorem

The only prime Dudeney number is $17$:

Proof

We have that:

\(\ds 17^3\)

\(=\)

\(\ds 4913\)

\(\ds 17\)

\(=\)

\(\ds 4 + 9 + 1 + 3\)

From Sequence of Dudeney Numbers, the complete set of positive integers with this property are:

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

Of these, only $17$ is prime.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $17$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$