Primitive integral solutions to $x^2+y^3=z^2$

All solutions of the Diophantine equation $x^2-y^2=z^3$ can be found in the book "Number Theory. Volume II: Analytic and Modern Tools" (Springer Science, 2007) by Henri Cohen. Below the relevant page from this book is reproduced.

As for general case of the Generalized Fermat equation, see https://www.staff.science.uu.nl/~beuke106/Fermatlectures.pdf The generalized Fermat equation, by Frits Beukers), http://homepages.warwick.ac.uk/~maseap/papers/bealconj.pdf (The Generalized Fermat Equation, by Michael Bennett, Preda Mihailescu and Samir Siksek) and http://people.math.sfu.ca/~ichen/pub/BeChDaYa.pdf (Generalized Fermat equation: a miscellany, by M.A. Bennett et al.).

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 $\ \gcd(s\ t) = 1$.