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

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# 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.

## Mathematics Subject Classification

11A63*no label found*

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