# Elementary Number Theory Chapter 3.3 Page 54, Why choose "$N = 4q_{1}q_{2} \cdots q_{s} - 1$"?(David M. Burton's 7th Edition)

Why does the author choose $N = 4q_{1}q_{2} \cdots q_{s} - 1$ when proving there are an infinite number of primes of the form $4n + 3$.

## Table of Contents

## Background

This question arises from David M. Burton's proof for "there are an infinite number of primes of the form $4n + 3$".

## Explanation

Why does the author choose $N$ to be $4q_{1}q_{2} \cdots q_{s} - 1$?

Because this can be written in the form of $4n + 3$. Also, it plays an important role when it leads to the contradiction that $r_{i} \mid 1$ as it has the "$-1$" at the end.

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