Question 136
Let $A$ be the number of 4-tuples $(x, y, z, t)$ of positive integers smaller than $2006^{2006}$ such that
$$
x^{3}+y^{2}=z^{3}+t^{2}+1,
$$
and let $B$ the number of 4-tuples $(x, y, z, t)$ of positive integers smaller than $2006^{2006}$ such that
$$
x^{3}+y^{2}=z^{3}+t^{2}.
$$
Prove that $B\gt A$.
$$
x^{3}+y^{2}=z^{3}+t^{2}+1,
$$
and let $B$ the number of 4-tuples $(x, y, z, t)$ of positive integers smaller than $2006^{2006}$ such that
$$
x^{3}+y^{2}=z^{3}+t^{2}.
$$
Prove that $B\gt A$.