Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$3435 = 3^3 + 4^4 + 3^3 + 5^5$ 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 $n = 0$.)

share|cite|improve this question
up vote 1 down vote accepted

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 $n>0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.