# If $P_1P_2…P_{n_0} + 1$ is prime and $2$ divides $P_2P_3…P_{n_0}+1$ is $(P_2P_3…P_{n_0} +1)/2$ prime or composite or both?

If $P_1P_2\cdots P_{n_0} + 1$ is prime, and 2 divides $P_2P_3\cdots P_{n_0}+1$, is $(P_2P_3\cdots P_{n_0} +1)/2$ prime or composite or both?

Here $P_i = \{{ 2,3,5,7 \dots\}}$

Composite numbers are those that are not prime.

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Have you tried any experiments? $2\times3+1$ is prime, and $2$ divides $3+1$, so you could see what happens there, and then look at a few more, and then tell us if you find anything interesting. – Gerry Myerson Nov 24 '11 at 11:26

Examples :

$a)$

$2\cdot 3 \cdot 5 +1=31$ - prime number

$\frac{3\cdot 5+1}{2}=8$ - composite number

$b)$

$2\cdot 3 \cdot 5 \cdot 7 +1=211$ - prime number

$\frac{3 \cdot 5 \cdot 7+1}{2}=53$ -prime number

So,we may conclude that $\frac{p_2\cdot p_3 \cdots p_n+1}{2}$ can be both prime and composite.

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But not at the same time. – Gerry Myerson Nov 24 '11 at 11:53
@GerryMyerson,I think that O.P. knows that number cannot be prime and composite at the same time... – pedja Nov 24 '11 at 12:03
yes, I know that a number cannot be prime and composite at same time. And thanks for the answer :) – titusfx Nov 24 '11 at 12:23
@Mauricio,You are welcome.. – pedja Nov 24 '11 at 12:39
Well, pedja, I just felt that as long as we were not leaving any work for OP to do, we might as well be thorough about it. – Gerry Myerson Nov 25 '11 at 0:09