Gaps between primes I recently watched a video about the recent breakthrough involving the gaps between primes.  I have an idea that I'm sure is wrong, but I don't know why.


*

*If you take the product of all prime numbers up to a certain number and call it x, won't x-1 and x+1 always be primes? 

*And since they always differ by 2, doesn't that make there an infinite number of primes that differ by 2?


Once again, I know that I'm wrong, but I would like to know why.

Say $p_k$ is the largest prime $\le$ some number $n$. Then take $\displaystyle x=p_1p_2\cdots p_k$ Now, of course $x-1$ and $x+1$ are not divisible by any of  $p_i,\ 1\le i\le k$. But there may be $p_k<p_i< x-1$ and $p_k<p_j<x+1$ such that $p_i|x-1$ and $p_j|x+1$. Just look at the example given by @DanielFischer to appreciate this fact.
