# Is the following always prime?

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.

• can you please share your program? Aug 12, 2019 at 19:29
• Note $\ v_k = t_k - 2\cdot 3\cdots p_k\,$ is coprime to $\,2,3,\ldots, p_k\ \$ Aug 12, 2019 at 19:36
• As @AccidentalFourierTransform notes, this sequence has received some attention. For that matter, easy plausibility/probabilistic heuristic arguments around the Prime Number Theorem strongly suggest that it is asymptotically true. E.g., maybe RH implies it's asymptotically true. Surely less than RH would give something, etc. Aug 12, 2019 at 20:09
• one plus any prime give an even number which can be written as the sum of two primes smaller than the initial prime.moving the primes before the initial prime gives your relation. unless one proves opposite. Aug 12, 2019 at 20:12
• This is known as Fortune's conjecture
– Sil
Aug 12, 2019 at 20:19