00:01
Okay, so here we can assume the contrary that there's going to only be finitely many primes that are congruent to 3 mod 4.
00:08
So then we can list the primes.
00:10
We're going to have a set and a finite set.
00:13
We can have p1, p2, and so on, up to, let's say, p sub k.
00:21
And then we can consider the integer n, which is going to be equal to 4 times p1, p2, and so on times pk.
00:30
We can note that n is going to be greater than one.
00:33
Now we observe that n is going to be congruent, right? we're going to have n here being equal to, again, n is equal to 4 times p1 times p2 and so on to times pk.
00:52
So again, we note here that n is going to be congruent to 3 mod 4.
01:03
And since n minus 3 is equal to, 4 times p1, p2, and so on, times pk minus 4, which is going to be a multiple of 4.
01:13
So our claims now is that, well, none of these primes, p1 to pk, is going to be a divisor of n...