Consider an integer of the form $8n +7$. Show that it cannot be expressed as a sum of three integer squares.
Hints are welcome. If you wish to post an answer, please post a hint as your answer, especially some fundamental concepts in elementary number theory / abstract algebra that might be relevant.
My work, so far:
As noted by carmichael561 (please see his hints below), the problem makes sense only for $n \in \mathbb{N}$.
Suppose, for contradiction, that
$$8n+7 = a^2 + b^2 + c^2$$
for $a,b,c \in \mathbb{Z}$.
We can rewrite the equation as $$8n = a^2 + b^2 + c^2 -7$$
Now since $8$ divides the LHS, it also divides the RHS. In particular,
the LHS is a number that is congruent to $0$ mod($8)$ but the LHS is congruent to $7$ mod($8$), which is a contradiction.
Hint: what are the squares mod $8$?