Q:

A small pattern in the primes. What can we do to continue the pattern ?

The pattern seems to have to do with divisors of the number $n$ or $n-1$.
$$
\begin{matrix}
p(1) & = 2 \\
p(2) & = 3 \\
p(3) & = 5 \\
p(4) & = 7 & = p(3) + (p(3) - p(2))\\
p(5) & = 11 & = p(4) + (p(2)^2 - p(3)) \\
p(6) & = 13 & = p(5) + (p(5) - p(2)^2) \\
p(7) & = 17 & = p(6) + (p(2)p(3) - p(5)) \\
p(8) & = 19 & = p(7) + (p(7) - p(2)p(3)) \\
p(9) & = 23 & = p(8) + (p(2)p(4) - p(7))\\
p(10) & = 29 & = p(9) + (p(3)^2 - p(2)p(4)) \\
p(11) & = \text{here the rule mysteriously ends}
\end{matrix}
$$
Any idea of how to generalize the rule so that it extends further than $p(11)$, where $p(n)$ is the $n$th prime number in $\Bbb{Z}$. 

A:

I'm not quite sure what rule you want here, that $p(n) = p(n - 1) + (\prod p(i) - \prod p(j))$? I think this would just follow from the fundamental theorem of arithemtic: $p(n) - p(n - 1)$ can be expressed as the difference between two numbers less than $p(n)$ (not proving this but feels obvious), which both then factor.

