proof that the set of sum-product numbers in base 10 is finite

This was first proven by David Wilson, a major contributor to Sloane’s Online Encyclopedia of Integer Sequences.

First, Wilson proved that ${10}^{m-1}\le n$ (where $m$ is the number of digits of $n$) and that

$$\sum _{i=1}^m{d}_i\le 9m$$

and

$$\prod _{i=1}^m{d}_i\le 9^m$$

. The only way to fulfill the inequality ${10}^{m-1}\le 9^{m}9m$ is for $m\le 84$.

Thus, a base 10 sum-product number can’t have more than 84 digits. From the first ${10}^{84}$ integers, we can discard all those integers with 0’s in their decimal representation. We can further eliminate those integers whose product of digits is not of the form $2^i{3}^j{7}^k$ or $3^i{5}^j{7}^k$.

Having thus reduced the number of integers to consider, a brute force search by computer yields the finite set of sum-product numbers in base 10: 0, 1, 135 and 144.