Problem Archive

$(x,y)$ is called a nested radical pair if $x$ and $y$ are non-zero integers such that $\dfrac{x}{y}$ is not a cube of a rational number, and there exist integers $a$, $b$ and $c$ such that:

$$\sqrt{\sqrt[3]{x}+\sqrt[3]{y}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$

For example, both $(-4,125)$ and $(5,5324)$ are nested radical pairs:

$$ \begin{align*} \begin{split} \sqrt{\sqrt[3]{-4}+\sqrt[3]{125}} &= \sqrt[3]{-1}+\sqrt[3]{2}+\sqrt[3]{4}\\ \sqrt{\sqrt[3]{5}+\sqrt[3]{5324}} &= \sqrt[3]{-2}+\sqrt[3]{20}+\sqrt[3]{25}\\ \end{split} \end{align*} $$

Let $H(N)$ be the sum of $|x|+|y|$ for all the nested radical pairs $(x, y)$ where $|x| \leq |y|\leq N$.
For example, $H(10^3)=2535$.

Find $H(10^{15})$. Give your answer modulo $1031^3+2$.