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

Background

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

Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)

All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 3)

RanblogRan

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 $3k +2$.