Euclid Mullin Sequence

Question

Consider the Sequence as follows.

Let $a_1 = 2$, $a_n$ be the largest prime divisor of $P_n = 1 + {\prod_{i = 1}^{n - 1} a_{i}} $

Then we obtain a sequence of prime numbers

How do you show that 5 is never in the sequence?

OK, I am not quite understanding the last paragraph here.

Specifically

1. I'm not sure what $(P_n, 6)$ mean

2. how was $P_{n_0} = 5^k$ reached

The book is The Development of Prime Number Theory by W. Narkiewicz.

Answer 1

1. $(P_n,6)$ is a shorthand for $\gcd(P_n,6)$. So $(P_n,6)=1$ is just saying that neither $2$ nor $3$ divides $P_n$.

2. $P_{n_0}=5^k$ is deduced from the fact that $P_{n_0}$ is not divisible by $2$ or $3$, yet its largest prime factor is supposed to be $5$.