91Ó°ÊÓ

Prove that there are infinitely many primes congruent to \(1 \mathrm{mod} 4,\) by filling in the details of the following outline: (1) Prove: if an odd prime \(p\) and \(n \in \mathbb{Z}\) are such that \(n^{2} \equiv-1(\bmod p)\), then \(4 \mid \phi(p)\) [use a theorem in this section]. Why is it necessary to exclude the even prime here? (2) Therefore for \(n \in \mathbb{Z}\), the odd prime divisors of \(n^{2}+1\) are congruent to 1 mod 4 . (3) Now assume for contradiction's sake that there are only finitely many primes \(p_{1}, \ldots, p_{n}\) congruent to \(1 \bmod 4\) and consider the number \(\left(2 p_{1} \cdots p_{n}\right)^{2}+1\) Apply the previous step....