Prove that there are an infinite number of primes of the for

Prove that there are an infinite number of primes of the form 4n-1.

Solution

Suppose there is a finite number of prime numbers of the form 4n1.
Let there be k of them: p1,p2,…,pk.
Let S={p1,p2,…,pk}.
Let N be constructed as:
N=4i=1kpi1
If N is a prime number, then it is of the form 4n1.
But then we have that NS, which means that S is not the complete set of prime numbers of the form 4n1.
Therefore N must be composite.
Suppose all the prime factors of N are of the form 4n+1.
Then from Product of Integers of form 4n+1 it follows that N itself is of the form 4n+1.
Therefore N must have at least one prime factor p of the form 4n1.
Suppose pS.
We have that:
p4i=1kpi
and so:
p4i=1kpip
So:
N=qp+(p1)
where:
q=4i=1kpipp
So applying the Euclidean Algorithm:
gcd{N,p} = gcd{p1,p}   = gcd{p1,1} = 1              
So if pS, then it cannot be a divisor of N.
Therefore there must be a prime numbers of the form 4n1 which is not in S.
Therefore S is not the complete set of prime numbers of the form 4n1.
Therefore the assumption that S is finite was false.
Hence the result.