Proving that an expression can never be a perfect square

Question

Is it possible to prove or disprove by modular arithmetic that $$f=\frac{(19(8m\pm1)+1)^2+(22(8m\pm1)+1)^2}{9}$$ can never be a perfect square ($m$ is any integer)? More generally, can a condition be derived on some $$f=\frac{(t(8m\pm1)+1)^2+(s(8m\pm1)+1)^2}{(s-t)^2}$$ to be an odd perfect square? ($t$ is always odd)