proof of infinite Primes of the Form 4n-1

Question

would like help on the proof, I have most of it but got stuck in one part. my proof is: lets assume there are finite number of primes in the form 4n-1. let P be their product, define N=4P-1. so none of the primes of the form 4n-1 divide N yet there is a prime that divide N, say q.

I know I need to show that q is of the form 4n-1 but have no idea how to prove it, would greatly appreciate your help

The product of two primes of the form $4n+1$ is again of the form $4n+1,$ so it can't be the case that every prime dividing $4P-1$ is of the form $4n+1.$ — saulspatz, Mar 07 '19 at 19:21
How does it help me? sorry for the lack of knowledge but i'm just starting to learn number theory — Idan Daniel, Mar 07 '19 at 19:23
Well, $P-1$ is divisible by an odd prime, and if not every prime dividing it is of the form $4n+1$ at least one must be of the form $4n-1$. — saulspatz, Mar 07 '19 at 19:24
saulspatz, why must it be of the form 4n+1 or 4n-1? can't there be other forms? — Idan Daniel, Mar 07 '19 at 19:27

Servaes · Accepted Answer · 2019-03-07T19:29:49.097

2

Suppose toward a contradiction that there is no prime of the form $4n-1$ that divides $N=4P-1$. Then every prime that divides $N=4P-1$ is of the form $4n+1$. But then the product of these primes is also of the form $4n+1$, contradicting the fact that their product is $N=4P-1$. Hence there exists a prime of the form $4n-1$ that divides $N=4P-1$.

edited Mar 07 '19 at 19:29

answered Mar 07 '19 at 19:26

Servaes

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Okay, I wasn't familiar that every prime must be of the form 4n+1 or 4n-1. why is it true? – Idan Daniel Mar 07 '19 at 19:29
@IdanDaniel Because every number is of the form $4n$, $4n+1$, $4n+2$ or $4n+3$. – Servaes Mar 07 '19 at 19:30
... and $4n$ or $4n+2$ can't be prime as they are divisible by $4$ and $2$ respectively. – learner Mar 07 '19 at 19:31
...except if they equal $2$. – Servaes Mar 07 '19 at 19:31
so what about the form 4n+3? are we saying it can always be replaced by the form 4n-1? – Idan Daniel Mar 07 '19 at 19:32
@IdanDaniel I do not understand what you are asking. – Servaes Mar 07 '19 at 19:32
I mean every prime number of the form 4n+3 can be written as 4t-1? – Idan Daniel Mar 07 '19 at 19:33
Yes, exactly; take $t=n+1$. – Servaes Mar 07 '19 at 19:34
Ok, sorry for I guess obvious questions. My professor never bothered to say every prime number must be of the two forms – Idan Daniel Mar 07 '19 at 19:36
@IdanDaniel and Servaes Please search before asking and answering FAQs. Rampant duplication is not good for the site. – Bill Dubuque Mar 07 '19 at 19:56
@BillDubuque As similar as these questions are (as are a dozen others), OP's specific question about this proof is not adressed in the supposed duplicate questions. Though the answers in the second question do seem helpful, at least. – Servaes Mar 07 '19 at 20:15
@Servaes Your answer is a dupe of tens (if not hundres) of prior answers. – Bill Dubuque Mar 07 '19 at 20:19
@BillDubuque Ok, I don't see it in the linked questions, but I also don't see a problem if it were. – Servaes Mar 07 '19 at 20:20

proof of infinite Primes of the Form 4n-1

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