Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$has solutions $(x, y) \in Z \times Z$.
Problem
Source: 2011 Romania JBMO TST 4.4
Tags: system of equations, Diophantine equation, diophantine, number theory