Prove that there are infinitely many primes of the form 6x-1.
I have been fighting this proof for an hour. I have no idea how to get started or on how to finish. Any help would be greatly appreciated.
All integers can be represented as:
Now, first we try to find the statements for which this is seemingly prime:
It's immediately clear that this is divisible by 6.
This has no immediate factors, MAY be prime.
It is immediately clear that it is divisible by two.
It is immediately clear that it is divisible by three.
Immediately clear that it is divisible by two.
It has no immediate factors. MAY be prime.
The only candidates for primacy are:
Therefore, all integers that are candidates for primacy are:
Since it has already been proven that there are infinitely many primes, then there are also infinitely many primes represented by each:
Thus, there are infinitely many primes represented by
I know it may not be as good as you need. But you can use it to build your own proof if you would like.
Indeed, suppose there's a finite number of primes of the form , let be all the primes of that form
This number must have a prime factor of the form , but it is not divisible by any prime of the form (by our assumption). This is absurd, therefore there have to be infnitely prime numbers of the form
We can generalize this result. Let . Suppose that all the (odd) primes are either . Then there are infinitely many primes . And the proof is similar. Suppose there are finitely many and define . Now any prime divisior of has form . It cannot be that all have form because then would have that form too. So it must mean that one of the prime factors of satisfies but then the LHS is divisible by but not the RHS.
By the Fundamental Theorem of Arithmetic, there must be a prime of form such that . But is not divisible by any prime of form by our assumption. This is a contradiction, thus, there must exist infinitely many primes of the form .
Such a proof should also hold for the case of primes in the form of too right?
All odd primes are either: .
If there are finitely many such primes of form it means we can form .
This number factors into odd primes. Say all these primes where of form . Then would be of form which is a contradiction. Thus, there exists , and odd prime divisor of form . But that forces which is a contradiction. Thus, there are infinitely many such primes.
And the same argument works for primes of form also.
Note: In case you are wondering proving that there are infinitely many primes of forms is more difficult.