$\displaystyle a,b,c \in \mathbb{N}^+, \quad d = 2a^2+1 = 3b^3+2 = 5c^5+3$
Find the smallest $\displaystyle d$ above
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).