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Problem 784

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Reciprocal Pairs

Let’s call a pair of positive integers $p$, $q$ ($p<q$) reciprocal, if there is a positive integer $r<p$ such that $r$ equals both the inverse of $p$ modulo $q$ and the inverse of $q$ modulo $p$.

For example, $(3,5)$ is one reciprocal pair for $r=2$.

Let $F(N)$ be the total sum of $p+q$ for all reciprocal pairs $(p,q)$ where $p\le N$.

$F(5)=59$ due to these four reciprocal pairs $(3,5)$, $(4,11)$, $(5,7)$ and $(5,19)$.

You are also given $F({10}^2)=697317$.

Find $F(2\cdot {10}^6)$.

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模倒数对

对于正整数$p$和$q$（满足$p<q$），若存在正整数$r<p$使得$r$同时是$p$同余$q$的逆元和$q$同余$p$的逆元，则称这两个正整数互为模倒数。

例如，$(3,5)$是一组模倒数对，其对应的$r=2$。

对所有满足$p\le N$的模倒数对$(p,q)$，记$F(N)$为所有$p+q$之和。

例如，$F(5)=59$，因为共有四组模倒数对$(3,5)$、$(4,11)$、$(5,7)$和$(5,19)$。

已知$F({10}^2)=697317$。

求$F(2\cdot {10}^6)$。

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