Problem 6

Find all positive integers

n

for which there exist nonnegative integers

a_1,\dots,a_n

such that

\sum_{i=1}^n2^{-a_i}=\sum_{i=1}^n i3^{-a_i}=1

Step 2 of 4: Pass from an odd length to the next even length

Previous step Next step

In plain words

One term can be replaced by two equal binary halves whose ternary indices add to three times the old index.

a'_{m+1}=a_{m+1}+1,\quad a'_{2m+2}=a_{m+1}+1

Detailed analysis

Suppose $(a_1,\ldots,a_{2m+1})$ is feasible. Define $a'_j=a_j$ except $a'_{m+1}=a_{m+1}+1$ and $a'_{2m+2}=a_{m+1}+1$ . The binary sum is unchanged because one term $2^{-a_{m+1}}$ becomes two copies of $2^{-(a_{m+1}+1)}$ . The ternary change is $-(m+1)3^{-a_{m+1}}+((m+1)+(2m+2))3^{-(a_{m+1}+1)}=0$ . Thus feasibility at $2m+1$ implies feasibility at $2m+2$ .