This should be written either in symbolic form:

or in words: "For all integers n, there does not exist an integer k such that

". In the first case, the notation "n : integer", as far as I know, comes from programming languages where it means "n is a variable of type

integer". It is also possible to write

where

is the standard notation for the set of integers. Equality should not be used to write this.

To prove this statement by contradiction, you assume that n is an integer and then assume the negation of what you have to prove, i.e., that there does exist an integer k such that

. One way to continue is to note that in this case n^2 is even and then to use an auxiliary fact that if a square of an integer x is even (this happens only when x itself is even), then x^2 is divisible by 4. From here, a contradiction can be obtained.

Forum rules say, "9. Start a new thread for a new question. Don't tag a new question onto an existing thread. Otherwise the thread can become confusing and difficult to follow." I guess, another reason for this is that unless a new thread is opened, people don't know that there is a new question; they think that whoever answered your first question is better equipped to answer any follow-ups. Also, we love when a whole thread can be marked as [SOLVED]