# Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q12

My Solution for "Let $p_{n}$ denote the $n$th prime number. For $n \geq 3$, prove that $p_{n+3}^{2} < p_{n}p_{n+1}p_{n+2}$. [Hint: Note that $p_{n+3}^{2} < 4p_{n+2}^2 <8p_{n+1}p_{n+2}$.]"

Background

All theorems, corollaries, and definitions listed in the book's order:

**I will only use theorems or facts that are proved before this question**. So you will not see that I quote theorems or facts from the later chapters.

## Question

Let $p_{n}$ denote the $n$th prime number. For $n \geq 3$, prove that $p_{n+3}^{2} < p_{n}p_{n+1}p_{n+2}$.

[Hint: Note that $p_{n+3}^{2} < 4p_{n+2}^2 <8p_{n+1}p_{n+2}$.]

## Solution

For this question, we need to use the theorem from Bertrand's Postulate mentioned in Chapter 3.2, $p_{n+1} < 2p_{n}$.

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