Primitive integral solutions to $x^2+y^3=z^2$ The Diophantine equation
$$x^2+y^3=z^2$$
has solutions $(\pm 1,2,\pm 3)$ and $(\pm 13,3,\pm 14)$.
Brown [Int. Math. Res. Not. IMRN 2012, no. 2, 423–436; MR2876388] states that "there are infinitely many parameterized solutions with $xyz \neq 0$ and $z^2 \neq 1$ when $n \leq 5$" to the equation
$$x^2 +y^3 = z^n,$$
but does not give a reference. I am looking for a reference that catalogs all primitive solutions.
 A: 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 $\ \gcd(s\ t) = 1$.

A: 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.).

