 Cardano triplet

A Cardano triplet is a triple of positive integers $(a,b,c)$ satisfyingwith real cube roots. Put $u=\sqrt[3]{a+b\sqrt{c}}$ and $v=\sqrt[3]{a-b\sqrt{c}}$. Cubing $u+v=1$ gives $uv=(1-2a)/3$, and cubing that product yields the necessary integer identityIt is also sufficient. From this identity, the real cube root of $a^2-b^{2}c$ gives $uv=(1-2a)/3$. Therefore $s=u+v$ satisfiesSince $a\ge 1$, the quadratic polynomial has negative discriminant $1-8a$ and no real roots. Hence $s=1$. This checks that cubing introduced no extraneous real solutions.

Reducing the integer identity modulo three forces $a\equiv 2(\mathrm{mod}3)$. Writing $a=3k-1$ for a positive integer $k$ then reduces it to $b^{2}c=k^2(8k-3)$. For example, $k=1$ gives $(a,b,c)=(2,1,5)$.