That the set of prime integers is infinite can be proved using the irrationality of $\pi$; see this wikipedia link. It analyzes the representation
$\tag 1 {\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}\times {\frac {5}{4}}\times {\frac {7}{8}}\times {\frac {11}{12}}\times {\frac {13}{12}}\times {\frac {17}{16}}\times {\frac {19}{20}}\times {\frac {23}{24}}\times {\frac {29}{28}}\times {\frac {31}{32}}\times \cdots }$
It 'seems only fair' that the same thing can be done using $e$:
Question 1: Has a proof that there are an infinite number of primes been constructed that uses the irrationality of $e$?
If the existence of such a proof is not available, then
Question 1-1: Is a proof available using any properties of $e$?
If again, no answers, then
Question 2: Can someone show that the set of prime integers is infinite using properties of Euler's constant?
My work
I found this representation of $e$,
$\tag 2 {\displaystyle e=1+{\cfrac {2}{1+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{18+{\cfrac {1}{22+{\cfrac {1}{26+\ddots \,}}}}}}}}}}}}}}.}$
Note that with $\text{(1)}$ there is a 'corresponding' sequence
$\tag 3 4,8,12,16,\dots$
and that with $\text{(2)}$ we can see
$\tag 4 6, 10,14,18,\dots$
I just find this interesting; perhaps it supplies a 'connection'.
We know that eventually every prime number will divide a term in the $\text{(4)}$ sequence.
If the only primes are contained in the finite set $\mathcal P = \{2,3,5, \dots, p\}$, then let $\beta = 2 \times 3 \times 5 \times \dots \times p$.
If we analyze $\text{(2)}$ using base $\beta$, the expansion of $e$ would eventually be repetitive or simply terminate.
(the statement in bold is an unproven bold statement)
We can give a proof based on the transcendentality of $e$:
Since $$\lim_{n\rightarrow\infty}\left(\prod_{i=1}^{n} p_i\right)^{1/p_n}=e,$$
if the set of prime numbers was finite then $e$ would be algebraic.