Q: A question in proof of result that there are infinitely many primes of form $8a-1$. I am self studying  number theory from David M Burton and I have a question in theorem on page 182 of 7 th edition whose image I am adding below. 

I am unable to understand why if all odd prime divisors are of form $8k+1$ then $N$ must be of form $8a+1$ . Why didn't the author considered even divisors of $N$. 

A: \begin{align*}
(8a+1)(8b+1) &= 64ab + 8a + 8b + 1  \\
    &= 8(8ab+a+b)+1  \text{,}
\end{align*}
so a product of any quantity of numbers having the form $8n+1$ is another number of the same form.
The proof is to show there is an odd prime of a certain form.  The only possible even prime divisor of $N$ is $2$ (appearing with power $1$), which is of no use in showing the existence of the desired odd prime.
