Pluperfect Digital Invariant has less than 61 Digits
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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): $61$