A warm-up:

$$ \left(\frac{s^n-t^n}2\right)^2 +\ (s\cdot t)^n\ =\ 
        \left(\frac{s^n+t^n}2\right)^2 $$

provides integer solutions for $\,\ s\equiv t \mod 2 $.

*(Sorry, I couldn't help it).*

 
>**REMARK**   More generally, going in the *abc* direction:
>
>$$ \left(\frac{s^m-t^n}2\right)^2 +\ s^m\cdot t^n\ =\ 
        \left(\frac{s^m+t^n}2\right)^2 $$
>
>for $\ s\equiv t\equiv 1 \mod 2,\ $ and $\ \gcd(s\ t) = 1$.