# Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q3

Find all pairs of primes $p$ and $q$ satisfying $p - q = 3$.

## Table of Contents

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

Find all pairs of primes $p$ and $q$ satisfying $p - q = 3$.

## Solution

If $q = 2$, $p = q +3 = 5$. This is the first pair.

For $q > 2$, we know $q$ is odd and $p = q + 3$ is an even integer. But this is impossible since both $p$ and $q$ are odd integers.

Therefore, $p = 5$ and $q = 2$ is the only pair.

**Read More:** All My Solutions for This Book

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