Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?
My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,….,p_k$
Consider $N = 4p_1p_2p_3…p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1…q_r$, where $q_i∈P$
Claim: At least one of the $q_i$'s is of the form $4n+3$:
Proof for my claim: $N$ is odd $\Rightarrow q_1,…,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$
If all $q_1,…q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$
Therefore, $N=q_1…q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.
Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$
$\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$
If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1…p_k + 3 \Rightarrow q_j=3$ Contradiction!
If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1…p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!
In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?
Regards
Best Answer
Suppose $n>1$ is an integer. We define $N=(n!)^2 +1$. Suppose $p$ is the smallest prime divisor of $N$. Since $N$ is odd, $p$ cannot be equal to $2$. It is clear that $p$ is bigger than $n$ (otherwise $p \mid 1$). If we show that $p$ is of the form $4k+1$ then we can repeat the procedure replacing $n$ with $p$ and we produce an infinite sequence of primes of the form $4k+1$.
We know that $p$ has the form $4k+1$ or $4k+3$. Since $p\mid N$ we have $$ (n!)^2 \equiv -1 \ \ (p) \ . $$ Therefore $$ (n!)^{p-1} \equiv (-1)^{ \frac{p-1}{2} } \ \ (p) \ . $$ Using Fermat's little Theorem we get $$ (-1)^{ \frac{p-1}{2} } \equiv 1 \ \ (p) \ . $$ If $p$ was of the form $4k+3$ then $\frac{p-1}{2} =2k+1$ is odd and therefore we obtain $-1 \equiv 1 \ \ (p)$ or $p \mid 2$ which is a contradiction since $p$ is odd.