Solve equation. sum of negatie powers of two equal to one. Diaphantite.

Question

Is the following correct? Let $\sum_{i=1}^n \frac{1}{2^{x_i}}=1$ where $x_i \in \mathbb{N}_0$ for $i \in \{1,\ldots,n\}$ than the only solutions is

$$x_i=n-1, \quad \forall i \in \{1,\ldots,n\}.$$

If so how to show it or disprove it? Thank you

Answer 1

Here is a nice proof. If all $x_i$ are different, multiply by the largest power $2^{x_i}$ to get an equation of the form $$2^{x_i}=1+\sum 2^{x_k-x_i}$$ and this is impossible, therefor at least two of the $x_i$ are equal. Now note that $$\frac{1}{2^k}+\frac{1}{2^k}=\frac{1}{2^{k-1}}$$ and use induction over $n$ to see that there is only one solution.

However the question needs to be correctly posed since

$$\frac{1}{4}+\frac{1}{4}+\frac{1}{2}=1$$