If \(\displaystyle 2a^2+1 = 3b^2+2\) then \(\displaystyle 2a^2\equiv 1\!\!\!\pmod3\), which is impossible. So no solutions there. But the remaining equations \(\displaystyle d = 2a^2+1 = 5c^5+3\) have at least one solution d = 163 (with a = 9 and c = 2).