Primes and proofs 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.
 A: Do not speak of "infinite primes" unless you mean "primes, each one of which, by itself, is infinite".  If you're talking about what Euclid proved, that can be expressed by speaking of "infinitely many primes".
Euclid wrote that if you take any finite set of primes and multiply them and then add 1, then the prime factors of the number you get are not among those you started with.  For example
$$
2\cdot3\cdot5\cdot7\cdot11\cdot13 + 1 = 59\cdot509.
$$
The primes we started with were 2, 3, 5, 7, 11, 13; the ones we got were 59 and 509.
Another example:
$$
5\cdot7+1 = 2\cdot2\cdot3\cdot3.
$$
The primes we started with were 5 and 7; the ones we got were 2 and 3.
Another example:
$$
3\cdot5 + 1 = 2\cdot2\cdot2\cdot2.
$$
The primes we started with were 3 and 5; the one we got was 2.
That's was Euclid wrote about: multiply finitely many primes; add 1; factor the result.  He said the primes you get will never be among the ones you started with; so you'll always end up with more primes than you started with.
The questions in your first paragraph may require some difficult original research, I think.
A: Guy writes $p\#$ for the product of the primes up to $p$, and discusses the primality of $p\#\pm1$ in problem A2 of Unsolved Problems In Number Theory. 
