I have in my notes the form of the integers as:
Now, I know that I have to use the division algorithim to prove the first form, and I can do this, but in the second form of an integer $4k$ isn't the "so on" redundant by the division alogirithim since there is no integer of the form $x=4k+4$ or does the "so on" simply imply that we can also have $5k, 5k+1....5k+4$ and $6k,6k+1...$?
I have a couple of other questions. I know how to prove the form of the square of any integer, however, in my proof, and any other proof, we assume $k=3q^2+2qr$ and I have also done a similar thing for proving the form of a cube. However, how is this valid since we have assumed the form of every integer to be $3q^2+2qr$, and then not implemented it in the for the form of the square?