Question: "Prove by contradiction that there exists no 'n' that is an element of the natural numbers, such that n^2 + 2 is divsible by 4"

My Solution:

By way of contradiction assume their exists 'n', such that n^2 + 2 is divisible by 4.

When n is even, such that n = 2p,

(2p)^2 + 2

= 4p^2 + 2

= 4k+ 2 which is not divisible by 4 (due to the remainder of 2)

When n is odd such that n = 2p+1

(2p+1)^2 + 2

= 4p^2 + 4p + 3

= 4(p^2+p) +3

4k+3, which is not divisible by 4 (due to the remainder of 3)

Therefore, this contradicts the original statement.

Hence, there exists no 'n' such that n^2 + 2 is divisible by 4

I am fairly new to the topic of proof so am not too confident yet. That's why I am asking if there are any flaws in my proof

Cheers