Arithmetic of integers

Every odd prime number has the form $4 k+1$ or $4 k-1$. Using the multiplicativity of the set $S$ from item 1, prove that the set of prime numbers of the form $4 k-1$ is infinite.

Prove that there are infinitely many prime numbers of the form $4 k+1$, relying on the following nontrivial statement.

If $n, m \in \mathbb {Z}$, $\gcd (n, m)=1$, and if $p$ is a prime number dividing $n^{2}+m^{2}$, then $p=4 k+1$.

If a natural number $n$ is divisible by exactly $r$ distinct prime numbers $p_{1}, \ldots , p_{r}$, then the number of integers less than $n$ and coprime to $n$ equals

The function $\varphi : \mathbb {N} \rightarrow \mathbb {N}$ is called Euler's function.

Verify the validity of the formula for the values $\varphi (n)$ for $n \leqslant 25$ and for $n=p^{m}$.

Using the binomial formula, prove by induction on $n$ that if $p$ is a prime number, then $n^{p}-n$ is divisible by $p$ for every $n \in \mathbb {Z}$.