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At the beginning I would like to ask if there are infinite prime numbers of the form:

$$\prod_{i=1}^{n} p_i + 1$$

where $p_i$ is the $i$-th prime number; but after a google search I found that they are called Primorial primes and it is not known if there are infinite of them.

I'm not a mathematician and I'm wondering if there exist (one or more) infinite sequence of primes that can be represented using a math expression similar to the product above (or a sum) ?

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    $\begingroup$ There are an infinite number of primes of the form $n$. There are an infinite number of primes of the form $2n+1$. Less trivially, there are an infinite number of primes of the form $an+b$, provided $\gcd(a,b)=1$. Beyond that, much more is unknown than known. $\endgroup$ Feb 21 '13 at 23:16
  • $\begingroup$ The fact Gerry mentions is a rather profound, beautiful result known as Dirichlet's Theorem on primes in arithmetic progressions...google it. $\endgroup$
    – DonAntonio
    Feb 21 '13 at 23:21
  • $\begingroup$ @GerryMyerson: disheartening :-( $\endgroup$
    – Vor
    Feb 21 '13 at 23:21
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    $\begingroup$ On the contrary --- look how much work there is for us to do! How much we have to look forward to! $\endgroup$ Feb 21 '13 at 23:35
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Let $$\theta=1.306377883863080690... $$ then $$f(n)=\lfloor \theta^{3^n} \rfloor $$ is an increasing sequence, always prime for all $n$.


The reason this works is that the primes are sufficiently close together that we can always add later digits to this number so it hits another prime, without changing its values for smaller $n$.


Source http://mathworld.wolfram.com/MillsConstant.html

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    $\begingroup$ It's actually not known that this is true! Mills proved that there is some constant, and it has now been proven that the constant can be taken as 1.3... under the (G)RH, but this has not been proved unconditionally to my knowledge. All that can be said is that only finitely many are composite with that 'natural' choice. $\endgroup$
    – Charles
    Feb 23 '13 at 3:12

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