Proof.
The proof is similar to the proof of Euclid's Theorem, but, for variety,
I will explain it in a slightly different way.
Suppose
are primes of the form
. Consider
the number
Then
![$ p_i \nmid N$](img29.png)
for any
![$ i$](img30.png)
. Moreover, not every prime
![$ p\mid N$](img31.png)
is
of the form
![$ 4x+1$](img32.png)
; if they all were, then
![$ N$](img33.png)
would also be of the
form
![$ 4x+1$](img32.png)
, which it is not.
Thus there is a
![$ p\mid N$](img31.png)
that is of the form
![$ 4x-1$](img25.png)
. Since
![$ p\not= p_i$](img34.png)
for any
![$ i$](img30.png)
, we have found another prime of the form
![$ 4x-1$](img25.png)
. We can repeat
this process indefinitely, so the set of primes of the form
![$ 4x-1$](img25.png)
is infinite.
Example 2.2
Set
![$ p_1=3$](img35.png)
,
![$ p_2=7$](img36.png)
. Then
is a prime of the form
![$ 4x-1$](img25.png)
.
Next
which is a again a prime of the form
![$ 4x-1$](img25.png)
.
Again:
This time
![$ 61$](img40.png)
is a prime, but it is of the form
![$ 4x+1 = 4\times 15+1$](img41.png)
.
However,
![$ 796751$](img42.png)
is prime and
![$ (796751-(-1))/4 = 199188$](img43.png)
.
We are unstoppable
This time the small prime,
![$ 5591$](img45.png)
, is of the form
![$ 4x-1$](img25.png)
and the large
one is of the form
![$ 4x+1$](img32.png)
.
Etc!