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Problem 835

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Supernatural Triangles

A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers.

Let $S(N)$ be the sum of the perimeters of all distinct supernatural triangles with perimeters less than or equal to $N$.

For example, $S(100)=258$ and $S(10000)=172004$.

Find $S({10}^{{10}^{10}})$. Give your answer modulo $1234567891$.

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超自然三角形

若一个毕达哥拉斯三角形（译注：三边长均为正整数的直角三角形）的三边中有两边长是连续整数，则称其为超自然三角形。

考虑所有不同的、周长小于等于$N$的超自然三角形，记其周长之和为$S(N)$。

例如，$S(100)=258$，$S(10000)=172004$。

求$S({10}^{{10}^{10}})$，并将你的答案对$1234567891$取余。

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