1) Are there infinitely many primes of the form $a_n$? if $p_1 = 2 < p_2 = 3 <\cdots$ is the sequence of primes then are there infinitely many $n$ for which $p_1p_2\dots p_n + 1$ is prime? For which $p_1p_2\dots p_n - 1$ is prime? Let us determine an infinite sequence of primes by starting with prime $q_1$ and then letting $q_n$ be some prime divisor of $q_1 q_2 \cdots q_{n-1} +1$. Can this be arranged so that the sequence $q_1,q_2,\ldots$ is a re-arrangement of the set of all primes? what if $q_n$ is the smallest prime divisor of $q_1 q_2\cdots q_{n-1} +1$
2) Also, as per Euclid proof for primes, $3 (5) + 1 = 16$ is not prime. How you can say the Euclid proof is great for infinite primes?
Generalize the both questions.