Let f(x) = [ n + p sin x], `x in (0,pi), n in Z`, p is a prime number and [x] = the greatest integer less than or equal to x. The number of points at which f(x) is not not differentiable is :

Answer

To solve the problem, we need to analyze the function \( f(x) = [n + p \sin x] \) where \( x \in (0, \pi) \), \( n \in \mathbb{Z} \), and \( p \) is a prime number. The notation \( [x] \) denotes the greatest integer less than or equal to \( x \).

### Step-by-Step Solution:

1. **Understanding the Range of \( \sin x \)**:
- The function \( \sin x \) varies from \( 0 \) to \( 1 \) for \( x \in (0, \pi) \).
- Therefore, \( p \sin x \) will vary from \( 0 \) to \( p \).

2. **Analyzing \( n + p \sin x \)**:
- Since \( n \) is an integer and \( p \sin x \) varies continuously from \( 0 \) to \( p \), the expression \( n + p \sin x \) will vary from \( n \) to \( n + p \).
- The greatest integer function \( [n + p \sin x] \) will change its value at integer points.

3. **Identifying Points of Non-Differentiability**:
- The function \( f(x) \) will not be differentiable at points where \( n + p \sin x \) is an integer.
- Let \( k \) be an integer such that \( n \leq k < n + p \). The condition for non-differentiability is:
\[ n + p \sin x = k \implies p \sin x = k - n \implies \sin x = \frac{k - n}{p} \]
- For \( \sin x \) to be valid, \( \frac{k - n}{p} \) must lie within \( [0, 1] \). Thus, we have:
\[ 0 \leq k - n < p \]
- This implies:
\[ n \leq k < n + p \]

4. **Counting Valid Integer Values of \( k \)**:
- The integers \( k \) can take values from \( n \) to \( n + p - 1 \). This gives us \( p \) possible values for \( k \).

5. **Finding Corresponding \( x \) Values**:
- For each integer \( k \), there are two corresponding \( x \) values in the interval \( (0, \pi) \):
- \( x = \sin^{-1}\left(\frac{k - n}{p}\right) \)
- \( x = \pi - \sin^{-1}\left(\frac{k - n}{p}\right) \)
- Thus, for each \( k \), we have two points where \( f(x) \) is not differentiable.

6. **Total Points of Non-Differentiability**:
- Since there are \( p \) integers \( k \) from \( n \) to \( n + p - 1 \), and each \( k \) contributes 2 points, the total number of points at which \( f(x) \) is not differentiable is:
\[ 2p \]

### Conclusion:
The number of points at which the function \( f(x) \) is not differentiable is \( 2p \).