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880 - Nested Radicals

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Problem 880: Nested Radicals

$(x, y)$ 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 $1031^3 + 2$.