# Imperfect digit-to-digit invariants in Base $10$

$3435 = 3^3 + 4^4 + 3^3 + 5^5$ is an example of a perfect digit-to-digit invariant.

Fact: The number of PDDIs is finite for any given base; in particular, for base $10$.

Question: Working over base $10$, fix an integer $n \in \mathbb{Z}$ and compute the sum $S_k$ as done above for $3435$ for every positive integer $k$. Must the union $\{k: S_k - n = k\} \cup \{k: S_k + n = k\}$ be finite?

(Note that the fact above says this union is finite when $n = 0$.)

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Let $d$ be the number of digits of $k \in \mathbb N$. Then $k \ge 10^{d-1}$, while $S_k \le 9^9 d$. Notice that $S_k / k \to 0$ very quickly as $d$ increases, so for large $d$ it is impossible for $S_k$ to be remotely close to $k$. That's why there are only finitely many PDDIs, and also why there are only finitely many occurrences of any fixed value of $k - S_k$. For that matter, even the much larger set $\{k \in \mathbb N : S_k > k/n \}$ is finite for a given $n>0$.