MathLabs

Problem 3

Determine all positive integers kk for which there exist a positive integer mm and a set SS of positive integers such that any integer n>mn>m can be written as a sum of distinct elements of SS in exactly kk ways.
Step 4 of 5: Structure of S
S=T∪{x,2x,4x,8x,…},T finiteS=T\cup\{x,2x,4x,8x,\ldots\},\quad T\text{ finite}
Detailed analysis

The lemma shows that beyond some point SS consists of a geometric tail with ratio 22: S=T∪{x,2x,4x,…}S=T\cup\{x,2x,4x,\ldots\} for a finite set TT and a positive integer xx.