Proof.
Suppose
![$ p_1, p_2, \ldots, p_n$](img239.png)
are distinct primes of the form
![$ 4x-1$](img282.png)
. Consider
the number
Then
![$ p_i \nmid N$](img285.png)
for any
![$ i$](img223.png)
. Moreover, not every prime
![$ p\mid N$](img286.png)
is of the form
![$ 4x+1$](img287.png)
; if they all were, then
![$ N$](img288.png)
would be of the form
![$ 4x+1$](img287.png)
. Thus there is a
![$ p\mid N$](img286.png)
that is of the form
![$ 4x-1$](img282.png)
. Since
![$ p\not= p_i$](img289.png)
for any
![$ i$](img223.png)
, we have found a new prime of the form
![$ 4x-1$](img282.png)
. We can repeat this process indefinitely, so the set of primes
of the form
![$ 4x-1$](img282.png)
cannot be finite.
Example 1.2
Set
![$ p_1=3$](img290.png)
,
![$ p_2=7$](img291.png)
. Then
is a prime of the form
![$ 4x-1$](img282.png)
. Next
which is again a prime of the form
![$ 4x-1$](img282.png)
.
Again:
This time
![$ 61$](img295.png)
is a prime, but it is of the form
![$ 4x+1 = 4\cdot 15+1$](img296.png)
.
However,
![$ 796751$](img297.png)
is prime and
![$ 796751 = 4\cdot 199188 - 1$](img298.png)
.
We are unstoppable:
This time the small prime,
![$ 5591$](img300.png)
, is of the form
![$ 4x-1$](img282.png)
and the large
one is of the form
![$ 4x+1$](img287.png)
.