The Diophantine Equations x2± y4=±z6 and x2+y8= z3

In this article we determine all solutions to the equation x p +y q =z r , (p,q,r)∈{(2,4,6), (2,6,4), (4,6,2), (2,8,3)} in coprime integers x,y,z. First we determine a set of curves of genus 2, such that every solution corresponds to a rational point on one of these curves. Then we determine the rational points on these curves using either covers of rank 0 elliptic curves or a method known as effective Chabauty which works if the Mordell–Weil rank of the jacobian is smaller than the dimension.