# Thread: Inequality with four consecutive prime numbers

1. ## Inequality with four consecutive prime numbers

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...

2. ## Re: Inequality with four consecutive prime numbers

As 8 does not depend on n, it would be sufficient to show pn>8 for n>=3.
This is wrong, but the few exceptions can be checked manually.

3. ## Re: Inequality with four consecutive prime numbers

$\displaystyle p_{n+3} < 8p_{n+1}p_{n+2}$. Here, if you can show that $\displaystyle 8 < p_n$, you are done. This is not true for n = 3 or n = 4, but just like mfb said, you can check those cases separately.

4. ## Re: Inequality with four consecutive prime numbers

Thank you both for your kind replies.
That's exactly like you say. Of course, it's perfectly ok to check these two exceptions manually, however, I am wondering how to avoid it. Can you see any way to finish it more 'formally'?