I assume that the $$a^6 + 4 b^3 = c^6$$
has no solution in integers. I think this can be solved trivially, but no success so far. I tried to treat this as a $$(a^3)^2 + 4 b^3 = (c^3)^2 \\ (a^2)^3 + 4 b^3 = (c^2)^3$$ But no success.
Could you please help to find any non-trivial integer solution or prove that there is no such.