# 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 $10^{m-1}\leq n$ (where $m$ is the number of digits of $n$) and that

 $\sum_{i=1}^{m}d_{i}\leq 9m$

and

 $\prod_{i=1}^{m}d_{i}\leq 9^{m}$

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

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.

Title proof that the set of sum-product numbers in base 10 is finite ProofThatTheSetOfSumproductNumbersInBase10IsFinite 2013-03-22 15:46:58 2013-03-22 15:46:58 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Proof msc 11A63 Proof that the set of sum-product numbers in decimal is finite