Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
To find the least positive integer that cannot be represented as $\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$, we need to analyze the form of the expression and the divisibility properties of powers of 2.

1. **Understanding the Expression**:
   The expression $\frac{2^a - 2^b}{2^c - 2^d}$ can be simplified by factoring out powers of 2:
   \[
   \frac{2^a - 2^b}{2^c - 2^d} = \frac{2^b(2^{a-b} - 1)}{2^d(2^{c-d} - 1)}
   \]
   This reduces to:
   \[
   \frac{2^{a-b} - 1}{2^{c-d} - 1}
   \]
   Let $x = a - b$ and $y = c - d$. The expression becomes:
   \[
   \frac{2^x - 1}{2^y - 1}
   \]

2. **Divisibility Condition**:
   For $\frac{2^x - 1}{2^y - 1}$ to be an integer, $2^y - 1$ must divide $2^x - 1$. This is true if and only if $y$ divides $x$ (i.e., $y \mid x$).

3. **Generating Representable Numbers**:
   We need to generate numbers of the form $2^k(1 + 2^\ell + 2^{2\ell} + \cdots + 2^{m\ell})$ where $k, \ell, m$ are non-negative integers. This is based on the fact that $2^p - 1 \mid 2^q - 1$ if and only if $p \mid q$.

4. **Checking Small Numbers**:
   We check small positive integers to see if they can be represented in the form above:
   - $1 = 2^0(1)$
   - $3 = 2^1(1 + 2^0)$
   - $5 = 2^2(1 + 2^0)$
   - $7 = 2^2(1 + 2^1)$
   - $9 = 2^3(1 + 2^0)$

5. **Finding the First Non-Representable Number**:
   We continue this process to find the first integer that cannot be represented in this form. The next integer to check is $11$:
   - $11$ cannot be written as $2^k(1 + 2^\ell + 2^{2\ell} + \cdots + 2^{m\ell})$ for any $k, \ell, m$.

Therefore, the least positive integer that cannot be represented as $\frac{2^a - 2^b}{2^c - 2^d}$ is $11$.

The final answer is $\boxed{11}$.