MathNet 2011 Question 86

Show that there is an infinite number of positive integers $t$ such that none of the equations $x^2 + y^6 = t$, $x^2 + y^6 = t + 1$, $x^2 - y^6 = t$, $x^2 - y^6 = t + 1$ has solutions $(x, y) \in \mathbb{Z} \times \mathbb{Z}$.