I think I proved something, but to be sure I'm right about it, I'm gonna let others trying to prove it (or disprove it) themselves. I'm not so good in all that math symbol thing, so I'm just going to pick a random symbol for all the things here. just go with it please.
let P(n) be the nth prime. for example P(1) = 2, P(2) = 3 , P(4) = 7....
now let Φ(n) = P(1) * P(2) * P(3) ... * P(n)
so what I think I proved is that both Φ(n)+1 and Φ(n) - 1 are primes (twin primes actually).
it's a simple proof, but I think I might be wrong here so I invite anyone who wants to, to prove or disprove it.
hint: you can easily prove that Φ(n) is not divisible by any prime numbers under P(n), but what about prime numbers that are bigger than P(n)?