Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation 
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers $n$.
1. **Assume the recurrence relation and initial conditions:**
   Given the recurrence relation for the sequence of prime numbers \( p_1, p_2, \ldots \):
   \[
   p_{n+2} = p_{n+1} + p_n + k
   \]
   for some integer \( k \) and for all positive integers \( n \).

2. **Consider the characteristic equation:**
   To solve the recurrence relation, we first consider the homogeneous part:
   \[
   p_{n+2} = p_{n+1} + p_n
   \]
   The characteristic equation for this homogeneous recurrence relation is:
   \[
   m^2 = m + 1
   \]
   Solving this quadratic equation, we get:
   \[
   m^2 - m - 1 = 0
   \]
   The roots of this equation are:
   \[
   m = \frac{1 \pm \sqrt{5}}{2}
   \]
   Therefore, the general solution to the homogeneous part is:
   \[
   p_n = A \left( \frac{1 + \sqrt{5}}{2} \right)^n + B \left( \frac{1 - \sqrt{5}}{2} \right)^n
   \]
   where \( A \) and \( B \) are constants.

3. **Find a particular solution:**
   We need to find a particular solution to the non-homogeneous recurrence relation:
   \[
   p_{n+2} = p_{n+1} + p_n + k
   \]
   A typical particular solution for a constant non-homogeneous term \( k \) is \( p_n = -k \).

4. **Combine the homogeneous and particular solutions:**
   The general solution to the non-homogeneous recurrence relation is the sum of the homogeneous solution and the particular solution:
   \[
   p_n = A \left( \frac{1 + \sqrt{5}}{2} \right)^n + B \left( \frac{1 - \sqrt{5}}{2} \right)^n - k
   \]

5. **Analyze the sequence modulo \( p_1 \):**
   If we take the recurrence relation modulo \( p_1 \), we get a periodic sequence. This implies that there will be another number in the sequence that is congruent to 0 modulo \( p_1 \). 

6. **Consider the behavior of the sequence:**
   If \( A \geq 0 \), the sequence eventually increases, and since the sequence consists of prime numbers, it cannot be fully prime. Similarly, if \( A \leq 0 \), the sequence will not be fully prime. Therefore, \( A = 0 \).

7. **Simplify the solution:**
   With \( A = 0 \), the solution simplifies to:
   \[
   p_n = B \left( \frac{1 - \sqrt{5}}{2} \right)^n - k
   \]
   Since all \( p_n \) are prime numbers and must be integers, \( B \) must also be 0. Therefore, the sequence simplifies to:
   \[
   p_n = -k
   \]
   Since \( p_n \) are prime numbers, \( k \) must be the negative of a prime number. Let \( k = -p \) for some prime \( p \). Then:
   \[
   p_n = p
   \]
   for all \( n \).

Conclusion:
The only sequence of prime numbers that satisfies the given recurrence relation is a constant sequence where each term is the same prime number \( p \), and \( k = -p \).

The final answer is \( \boxed{ p_n = p } \) for all \( n \) and \( k = -p \).