proof that the set of sum-product numbers in base 2 is finite

For any positive integer with 0’s in its binary representation, the equality


is false, or we can rewrite it as the inequality n>0. Obviously, for n=0 the equality given above is true, thus 0 is the only sum-product number in binary that has 0’s in its binary representation.

This leaves the Mersenne numbers (numbers whose significant digits are all 1’s), an infinite set but just as manageable. The Mersenne numbers are usually expressed as 2k-1, but for our purposes we express them as i=0k-12i, which shows that for n=2k-1, i=1mdi=k. Regardless of how many digits a Mersenne number has, it is obvious that i=1mdi=1 because of the multiplicative identityPlanetmathPlanetmath. Thus, for a Mersenne number, the right hand side of our equality above will be k. For any nonnegative k, the inequality k<2k-1 will always hold except of course for k=1.

This proves that 0 and 1 are the only sum-product numbers in binary.

Title proof that the set of sum-product numbers in base 2 is finite
Canonical name ProofThatTheSetOfSumproductNumbersInBase2IsFinite
Date of creation 2013-03-22 15:46:56
Last modified on 2013-03-22 15:46:56
Owner Mravinci (12996)
Last modified by Mravinci (12996)
Numerical id 6
Author Mravinci (12996)
Entry type Proof
Classification msc 11A63
Synonym Proof that the set of sum-product numbers in binary is finite