Hi,

I want to show the following:

Let $\displaystyle n \geq 3$, $\displaystyle p_n$ - n-th prime number. Then $\displaystyle p^2_{n+3} < p_n p_{n+1} p_{n+2}$.

My line of reasoning:

$\displaystyle p_{n+3} < 2 p_{n+2} \Rightarrow p^2_{n+3} < 4 p^2_{n+2}$

$\displaystyle 4 p^2_{n+2} < 4 p_{n+2} \cdot 2 p_{n+1}$

So:

$\displaystyle p^2_{n+3} < 8 p_{n+2} p_{n+1}$

What next? I guess, I have to use $\displaystyle p_n < 2^n$

I will be very grateful for any help...