# Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q20

My Solution for "If $p$ and $p^{2} + 8$ are both prime numbers, prove that $p^{3} + 4$ is also prime."

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

If $p$ and $p^{2} + 8$ are both prime numbers, prove that $p^{3} + 4$ is also prime.

## Solution

By Theorem 2.1 Division Algorithm, we can write $p$ in the form of $3k$, $3k + 1$ or $3k +2$.

## The rest is for Premium Members only

SubscribeAlready have an account? Log in