Establishing infinitely many primes of the form $4k+1$.

Question

Establish that there are infinitely many primes of the form $4k+1$.

I was studying primitive roots, and it was recently proven that the odd prime divisors $n^2 +1$ are all of the form of $4k+1$.

A proof in Burton's Elementary Number Theory assumes that there are finitely many primes of the form $4k+1$. The book lets $N = (2p_1...p_n)^2 +1$ ($p_i$ are the primes of form $4k+1$ then proceeds that this number would have a prime of the form $4k+1$ that is not $p_i$ ($i$ between $1$ and $n$) because $p_i$ does not divide 1, thus does not divide $N$.

I was wondering why you would use $N =(2p_1...p_n)^2+1$, doesn't letting $N=(p_1...p_n)^2 +1$ achieve the exact same thing?

Answer 1

For every integer $n$, every odd prime divisor $p$ of $n^2+1$ is of the form $p=4k+1$ for some integer $k$. It follows that for every even integer $n$, every prime divisor $p$ of $n^2+1$ is of the form $p=4k+1$ for some integer $k$.

Let $p_1,\ldots,p_m$ be a finite list of primes. Then every odd prime divisor of $$N=(p_1\cdot p_2\cdots p_m)^2+1,$$ is a prime number of the form $4k+1$, and is coprime to $p_1,\ldots,p_m$. This shows that if $N$ has an odd prime divisor, then the finite list $p_1,\ldots,p_m$ is incomplete. So it remains to show that $N$ has an odd prime divisor, i.e. that $N$ is not a power of $2$.

On the other hand, every prime divisor of $$M=(2p_1\cdot p_2\cdots p_m)^2+1,$$ is a prime number of the form $4k+1$, and is coprime to $p_1,\ldots,p_m$. This shows that the finite list $p_1,\ldots,p_m$ is incomplete. No further argument is needed.