Infinite Primes

The aim of this note is to try and make a list of all possible prove that has infinitely many primes type of question. This is an open note in the sense it is never complete

Theorem (Infinite Primes ).

There are infinite Primes of the form equivalently there are infinity many primes

Proof.
We can see this is a direct result of Fundamental Theorem of Arithmetic.
Let us assume there are finitely many primes with the larges prime being where represents prime.
Lets define such that

Claim 1.1.

Let be the smallest divisor of and is a prime

Lets prove this my contradiction if is not a primes such that this contradicts the fact that is smallest hence is a prime.
as none of
We assumed there are finitely many primes but now found a new prime which is not equal to any of the other prime this shows that our assumption is false and there infinitely many prime.

Note: Even though we said its a direct consequence of Fundamental Theorem of Arithmetic we haven't used it in our theorem but can be used to make a simpler Proof

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Theorem (Infinite primes of form

There are infinitely many primes of form

Proof.
Stack exchange proof

Let be primes of the form .
We prove that there are infinitely primes of form by of contradiction by assuming that there are finitely many primes of form
Let .
Let be a prime and .

this shows
Case I:

Case II:

This shows that q should be congruent to in both cases
If as is prime but

This contradicts the fact there for .
This contradicts the fact that there are finitely many primes of the form

There are infinitely many primes of the form

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Theorem (Infinite primes of form

There are infinitely many primes of form

Proof.
We prove this by contradiction lets assume there are finitely many primes of the form
let be prime of the form
Let

using Fundamental Theorem of Arithmetic we can write as

where are prime divisors of .
Note:
All of can't be of the form as then the product will also be of the form so at least one of be of the form but that contradict the fact that are the only primes of form .
There are infinitely many primes of the form

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Theorem (Infinite primes of form

There are infinitely many primes of form

Proof.
We prove this by contradiction lets assume there are finitely many primes of the form
let be prime of the form
Let

using Fundamental Theorem of Arithmetic we can write as

where are prime divisors of .
Note:
All of can't be of the form as then the product will also be of the form , so at least one of must be of the form . But that contradict the fact that are the only primes of form .
There are infinitely many primes of the form
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Theorem (Infinite primes of form

There are infinitely many primes of form

We prove this but contradiction, let assume that there are finitely many primes of the form .
Let be the highest prime of form .
let

Let be a prime such that then

From Fermat's Little Theorem

Note:
We have shown a new prime of form which is greater than this contradicts the fact that there are finitely many primes.
We have shown that there are infinitely many primes of form