Is the following always prime?

prime-numbers

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.


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.

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