Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$. Determine all the natural numbers $ p$ for which:

\[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\]

is a power with natural exponent of $ 2$.
1. First, we need to find the sum \( s_p = x_0 + x_1 + x_2 + \cdots + x_p \) where \( x_n = 2^n - 1 \).

2. We can write the sum explicitly:
   \[
   s_p = (2^0 - 1) + (2^1 - 1) + (2^2 - 1) + \cdots + (2^p - 1)
   \]

3. Simplify the sum by separating the terms:
   \[
   s_p = (2^0 + 2^1 + 2^2 + \cdots + 2^p) - (1 + 1 + 1 + \cdots + 1)
   \]

4. The first part is a geometric series with the sum:
   \[
   2^0 + 2^1 + 2^2 + \cdots + 2^p = 2^{p+1} - 1
   \]
   (using the formula for the sum of a geometric series: \( \sum_{k=0}^n ar^k = a \frac{r^{n+1} - 1}{r - 1} \) with \( a = 1 \) and \( r = 2 \)).

5. The second part is simply the sum of \( p+1 \) ones:
   \[
   1 + 1 + 1 + \cdots + 1 = p + 1
   \]

6. Combining these results, we get:
   \[
   s_p = (2^{p+1} - 1) - (p + 1)
   \]

7. Simplify the expression:
   \[
   s_p = 2^{p+1} - 1 - p - 1 = 2^{p+1} - (p + 2)
   \]

8. We need \( s_p \) to be a power of 2, say \( 2^k \). Therefore, we set up the equation:
   \[
   2^{p+1} - (p + 2) = 2^k
   \]

9. Rearrange the equation:
   \[
   2^{p+1} - 2^k = p + 2
   \]

10. Since \( 2^{p+1} \) and \( 2^k \) are powers of 2, their difference must be a power of 2. Let \( 2^{p+1} - 2^k = 2^m \) for some integer \( m \). Then:
    \[
    2^m = p + 2
    \]

11. We need to find \( p \) such that \( p + 2 \) is a power of 2. Let \( p + 2 = 2^m \). Then:
    \[
    p = 2^m - 2
    \]

12. Substitute \( p = 2^m - 2 \) back into the original equation to check if it satisfies the condition:
    \[
    s_p = 2^{(2^m - 2) + 1} - ((2^m - 2) + 2) = 2^{2^m - 1} - 2^m
    \]

13. We need \( s_p \) to be a power of 2. Let \( s_p = 2^n \). Then:
    \[
    2^{2^m - 1} - 2^m = 2^n
    \]

14. The only solution that satisfies this equation is when \( m = 1 \). For \( m = 1 \):
    \[
    p = 2^1 - 2 = 0
    \]

15. Check if \( p = 0 \) satisfies the condition:
    \[
    s_0 = x_0 = 2^0 - 1 = 0
    \]
    which is not a power of 2.

16. For \( m = 2 \):
    \[
    p = 2^2 - 2 = 2
    \]

17. Check if \( p = 2 \) satisfies the condition:
    \[
    s_2 = x_0 + x_1 + x_2 = (2^0 - 1) + (2^1 - 1) + (2^2 - 1) = 0 + 1 + 3 = 4 = 2^2
    \]
    which is a power of 2.

Therefore, the only natural number \( p \) for which \( s_p \) is a power of 2 is \( p = 2 \).

The final answer is \( \boxed{ p = 2 } \).