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 10m-1n (where m is the number of digits of n) and that

i=1mdi9m

and

i=1mdi9m

. The only way to fulfill the inequality 10m-19m9m is for m84.

Thus, a base 10 sum-product number can’t have more than 84 digits. From the first 1084 integers, we can discard all those integers with 0’s in their decimal representation. We can further eliminate those integers whose productPlanetmathPlanetmath of digits is not of the form 2i3j7k or 3i5j7k.

Having thus reduced the number of integers to consider, a brute force search by computer yields the finite setMathworldPlanetmath 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
Canonical name ProofThatTheSetOfSumproductNumbersInBase10IsFinite
Date of creation 2013-03-22 15:46:58
Last modified on 2013-03-22 15:46:58
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Proof
Classification msc 11A63
Synonym Proof that the set of sum-product numbers in decimal is finite