First of all I was wondering what results there are out there about the prime Euclid numbers? I would like to read up on any results that are out there.

Secondly I wanted someone to look at the following argument about the infinitude of the prime Euclid numbers.

According to Dirichlets theorem: If $a,b\in \mathbb{Z^{+}}$ such that $(a,b)=1$, then there are infinitely many prime numbers of the form $am+b$ where $m$ is a non-negative integer. Now, if we denote the $n$th primorial (i.e the product of the first $n$ primes) as $\displaystyle \prod\_{i=1}^{n} p\_{i}$ we define the Euclid numbers as $$E\_{n}={\displaystyle \prod\_{i=1}^{n} p\_{i}}+1.$$ Note that $$E\_{n}={\displaystyle \prod\_{i=1}^{n} p\_{i}}+1=2\cdot {\displaystyle \prod\_{i=2}^{n} p\_{i}}+1$$ where the first term $$2\cdot {\displaystyle \prod\_{i=2}^{n} p\_{i}}\equiv 2\pmod 4.$$ Thus $$E\_{n}\equiv 3\pmod 4.$$ Meaning in particular that $$\boxed{E\_{n}={\displaystyle \prod\_{i=1}^{n} p\_{i}}+1=4m+3}$$ for some non-negative integer $m$.

Since $(4,3)=1$, then by Dirichlets theorem we have that $4m+3$, where $m$ is a non-negative integer is a prime number infinitely often. And since the $RHS$ of the boxed equation above has this form, then does this imply that whatever is on the $LHS$ must also be a prime number infinitely often? In particular, does this mean that $E\_{n}$ is prime infinitely often?