Show that there are infinitely many primes of the form
4k - 1.
Alternatively, this is equivalent to showing that there are infinite number of primes congruent to 3 mod 4.
For now, assume by way of contradiction that there are only a finite number of primes satisfying . Let’s order them as
Using a similar argument made by Euclid on the proof of the infiniteness of primes, we now define a composite number
We first claim that
Now, the congruence class of or of the odds is partitioned between 1 and 3. Now, assuming that all prime factors of are congruent to 1 mod 4, then it will be the case that . Therefore, it must be the case that at least one of the prime divisors of , or that .
Finally, we show that .
Therefore, there cannot exists a prime of the form that is a divisor of , which leads to a contradiction as the above shows that must have some as its factor. Therefore, it cannot be the case that there are only a finite number of primes of the form .