Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q24

My Solution for "Determine all twin primes $p$ and $q = p + 2$ for which $pq - 2$ is also prime."

Ran
Ran


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)

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

Determine all twin primes $p$ and $q = p + 2$ for which $pq - 2$ is also prime.

Solution

When $p < 5$, $p = 3$ and $q = 5$ is the only solution.

When $p \geq 5$, as we mentioned in Chapter 3.1 Q4, $p$ takes one of the forms $6k + 1$ or $6k + 5$ for some integer $k$ since all other forms are composite.

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