all prime numbers except 2 can write in the form $4npm1$

Question

First I saw this, I was excited When I checked prime numbers and saw they are in this form. after checking more numbers I understood that every odd numbers can be written in this form then I showed it:
$4npm1=2(2n)pm1rightarrow 2kpm1 $ ( $k$ is even)
Here are my questions:
Question$1$: Why We should represent prime numbers in this form when it is equivalent to saying every prime numbers except $2$ are odd?
Question$2$: here we represented odd numbers with $4npm1$. It is maybe the first time I see kind of exchange in representation in mathematician. Please explain when we use such exchanges in numbers representation and is there resource to study about it or any generalization of this?

jonan · Answer

Somewhat related and a little bit more interesting:
Every prime except 2 and 3 is of the form: $6k pm 1$. Fun exercise.

James · Answer

For your 1st question, the reason mathematicians differentiate between primes of the form 4n+1 and 4n-1 is because a lot of results only hold for 4n+1 primes and a lot of results only hold for 4n-1 primes, particularly in modular arithmetic, such as the number of solutions to x^2 + y^2 = 1 in prime fields. Hope this helps!

Answered by James on December 1, 2021

Rhys Hughes · Answer

Well, if $p=4k$, then $4|p$ and so $p$ is not prime.
If $p=4k+2=2(2k+1)$ then $2|p$ and so $p$ is not prime unless $p=2$.
The only cases left are $p=4kpm 1$, and neither have a general factor.

lhf · Answer

Hint: Every integer can be written as $4n+r$, with $r in {0,1,2,3}$. This is Euclidean division by $4$ with remainder. Note that $4n+3 = 4(n+1)-1$. When $r in {0,2}$, the number $4n+r$ is even.

all prime numbers except 2 can write in the form $4npm1$

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