TITLE: Is the following always prime?
QUESTION [10 upvotes]: For a given $k$ define
$$s_k = 1 + \prod_{i=1}^k p_i$$
$$t_k = \text{NextPrime}(s_k)$$
$$v_k = t_k - s_k +1$$
Where $p_i$ is the $i$th prime number.
Conjecture: $v_k$ is prime
Example: 
$$k=3$$
$$s_3 = 1+(2)(3)(5)=31$$
$$t_3 = 37 $$
$$v_3 = 37-31 +1 = 7 \,\text{(prime)} $$
Tested numerically up to $k=400$ and it seems to hold.

REPLY [6 votes]: This is sequence A005235 on OEIS. It is conjectured to always be prime, although it is not known yet. Check the page for useful remarks and references.