Modular Arithmetic proof .

Question

Show that there are infinitely many positive integers which are not the sum of three squares. [Hint: what are the squares mod 8?] Investigate whether a similar argument, working mod 16, could give a similar result about four squares.

Answer 1

Expanding upon the already provided hint:

Every square is one of ______ modulo $8$.

Every square is one of $0,1,4$ modulo $8$. (Make sure you can explain why this is!)

So, the possible results of sums of three squares modulo $8$ would be:

$0+0+0, 0+0+1, 0+1+1, 1+1+1, 0+0+4, 0+1+4,\dots$. (Complete the list and simplify it to $0,1,2,\dots$ and make sure you can explain why it is)

Now, what do you notice about this list? Is there anything obvious that is missing from it? Can you describe those numbers?

What does the absence of a particular number from this list imply about numbers with that remainder modulo 8 and their ability to be expressed as a sum of three squares?