= No integer solutions of the Mordell equation with constant minus twenty-one
{title2=$y^2=x^3-21$}
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.
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