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Problem 880

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Nested Radicals

$(x,y)$ is called a nested radical pair if $x$ and $y$ are non-zero integers such that $x/y$ 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:
$$\begin{array}{r}\begin{array}{rl}\sqrt{\sqrt[3]{-4}+\sqrt[3]{125}}& =\sqrt[3]{-1}+\sqrt[3]{2}+\sqrt[3]{4}\\ \sqrt{\sqrt[3]{5}+\sqrt[3]{5324}}& =\sqrt[3]{-2}+\sqrt[3]{20}+\sqrt[3]{25}\end{array}\end{array}$$
Let $H(N)$ be the sum of $|x|+|y|$ for all the nested radical pairs $(x,y)$ where $|x|\le |y|\le N$.

For example, $H({10}^3)=2535$.

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

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嵌套根式

若非零整数$x$和$y$满足，$x/y$不是有理数的立方，且存在满足以下条件的整数$a$、$b$、$c$，则称$(x,y)$为嵌套根式对：
$$\sqrt{\sqrt[3]{x}+\sqrt[3]{y}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$
例如，$(-4,125)$和$(5,5324)$都是嵌套根式对：
$$\begin{array}{r}\begin{array}{rl}\sqrt{\sqrt[3]{-4}+\sqrt[3]{125}}& =\sqrt[3]{-1}+\sqrt[3]{2}+\sqrt[3]{4}\\ \sqrt{\sqrt[3]{5}+\sqrt[3]{5324}}& =\sqrt[3]{-2}+\sqrt[3]{20}+\sqrt[3]{25}\end{array}\end{array}$$
考虑所有满足$|x|\le |y|\le N$的嵌套根式对$(x,y)$，记$H(N)$为所有$|x|+|y|$的和。

例如，$H({10}^3)=2535$。

求$H({10}^{15})$，并对${1031}^3+2$取余作为你的答案。

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