Putnam 1983 B6

Problem B-6 Let $k$ be a positive integer, let $m=2^{k}+1$, and let $r \neq 1$ be a complex root of $z^{m}-1=0$. Prove that there exist polynomials $P(z)$ and $Q(z)$ with integer coefficients such that $$(P(r))^{2}+(Q(r))^{2}=-1$$