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Problem 817

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Digits in Squares

Define $m = M(n, d)$ to be the smallest positive integer such that when $m^2$ is written in base $n$ it includes the base $n$ digit $d$. For example, $M(10,7) = 24$ because if all the squares are written out in base $10$ the first time the digit $7$ occurs is in $24^2 = 576$. $M(11,10) = 19$ as $19^2 = 361=2A9_{11}$.

Find $\displaystyle \sum_{d = 1}^{10^5}M(p, p - d)$ where $p = 10^9 + 7$.

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平方中的数字

考虑所有$n$进制表示包含数字$d$的平方数$m^2$，并记其算术平方根的最小值为$m = M(n, d)$。例如，$M(10,7) = 24$，因为$10$进制下最小的包含数字$7$的平方数是$25^2=576$。类似可得$M(11,10) = 19$，因为$19^2 = 361=2A9_{11}$。

求$\displaystyle \sum_{d = 1}^{10^5}M(p, p - d)$，其中$p = 10^9 + 7$。

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