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Problem 942

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Mersenne’s Square Root

Given a natural number $q$, let $p = 2^q - 1$ be the $q$-th Mersenne number.

Let $R(q)$ be the minimal square root of $q$ modulo $p$, if one exists. In other words, $R(q)$ is the smallest positive integer $x$ such that $x^2 - q$ is divisible by $p$.

For example, $R(5)=6$ and $R(17)=47569$.

Find $R(74\ 207\ 281)$. Give your answer modulo $10^9 + 7$.

Note: $2^{74207281}-1$ is prime.

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梅森平方根

对于给定自然数$q$，记$p = 2^q - 1$为第$q$个梅森数。

记$R(q)$为$q$在模$p$同余意义下的最小平方根（若存在）。换句话说，$R(q)$是满足$x^2 - q$能被$p$整除的最小正整数$x$。

例如，$R(5)=6$，$R(17)=47569$。

求$R(74\ 207\ 281)$，并对$10^9 + 7$取余作为你的答案。

注意：$2^{74207281}-1$是质数。

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