SAUDI ARABIAN MATHEMATICAL COMPETITIONS

Take $x_1=y_1=-1$, and the following relations $$x_{k+1}=x_k-6y_k,y_{k+1}=x_k+y_k.$$ for $k=1,2,3,\dots$

One can check that $$x_{k+1}^2+6y_{k+1}^2=7\left(x_k^2+6y_k^2\right)$$ and $$x_k\equiv y_k\equiv (-1{)}^k(\mathrm{mod}7).$$ Hence, there exist the integer $x=x_k,y=y_k$ satisfy the given condition.