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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.

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    On (2), while $3\times 5 +1$ is not prime, any prime factor of it is neither $3$ nor $5$, which is the type of result you want for Euclid's proof when you multiply together all primes (assuming there are a finite number) and add $1$.2011-10-02
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    As Henry says, Euclid's proof is that $p_1\cdots p_n+1$ is divisible by none of $p_1,p_2,\dots,p_n$, therefore must have a prime factor not among them, hence no finite list of primes can contain them all. I don't know if it's known whether $p\#+1$ (see [primorial](http://en.wikipedia.org/wiki/Primorial)) is prime infinitely often. I'm not sure if we have theoretical machinery powerful enough to handle any of question (1). (Also, if you're [this Gadhi](http://math.stackexchange.com/users/14423/gandhi#qpage_2-qsort_votes) and you want to, you could request your accounts get merged.)2011-10-02
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    I recommend you to look at the Euclid-Mullin sequence, http://en.wikipedia.org/wiki/Euclid%E2%80%93Mullin_sequence There you'll see that the second part of pint 1) is not so trivial.2011-10-02
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    Ok. I got it. What about my first question?2011-10-02
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    The several parts that make up the first question are all currently unsolved. They are moderately well-known, and probably all difficult.2011-10-02
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    @Henry: Euclid's proof did not multiply together all primes (assuming there are a finite number). Rather, it considered an arbitrary finite set of primes, and multiplied them and added 1, and showed that the prime factors of the resulting number were not among the primes you started with.2011-10-02
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    What is $a_n$ in the first sentence?2011-10-03

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