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.
Thanks in advance.
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.
Thanks in advance.
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.