No integer solutions of the Mordell equation with constant minus twenty-one ($y^2=x^3-21$)

Parity and valuations at 3 and 7 make the ideals $(y+\sqrt{-21})$ and $(y-\sqrt{-21})$ coprime in any hypothetical solution. Their product is a cube, so the first is an ideal cube. The ideal class group has exponent two, forcing its cube root ideal to be principal. Cubing $a+b\sqrt{-21}$ would then give coefficient $3b(a^2-7b^2)$ on $\sqrt{-21}$, which cannot equal one. This proves the nonexistence of integer solutions without assuming unique element factorization.