Pluperfect Digital Invariant has less than 61 Digits

Theorem

Let $n \in \Z_{>0}$ be a pluperfect digital invariant.

Then $n$ has less than $61$ digits.

Proof

We have that:

$n \times 9^n < 10^\paren {n - 1}$

when $n > 60$.

So an $n$-digit integer, for $n > 60$, is always greater than the sum of the $n$th powers of its digits.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $61$