Question 61
Determine all sequences $p_1, p_2, p_3, \dots$ 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 $n$.
$$
p_{n+2} = p_{n+1} + p_n + k
$$
holds for all positive integers $n$.