is the multiplication of n consecutive prime numbers starting with 2 plus 1 prime? The question kinda tell everything for itself, 
let P(n) be the n-th prime number, is
$(\Pi_1^n P_n)+1$
prime ?
 A: No, consider $1 + 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19$. It's divisible by $347$ and $27953$.
A: > gp
                            GP/PARI CALCULATOR Version 2.7.3 (released)
                    i386 running darwin (x86-64/GMP-6.0.0 kernel) 64-bit version
       compiled: May 24 2015, Apple LLVM version 6.0 (clang-600.0.57) (based on LLVM 3.5svn)
                                      threading engine: single
                           (readline v6.3 enabled, extended help enabled)

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PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY
WHATSOEVER.

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parisize = 8000000, primelimit = 500000
? factor(2*3*5*7*11*13*17*19+1)
%1 =
[  347 1]

[27953 1]


Some people maintain that Euclid proved the infinitude of primes by contradiction, but actually he made a direct proof.
Some proofs by contradiction run as follows.

Suppose $p_1,p_2,\dots,p_n$ are all the primes. Then $N=p_1p_2\dotsm p_n+1$ is not divisible by any of the given primes, so it is prime. Contradiction.

However this doesn't prove that the product of consecutive primes plus one is prime.
