Problem 817

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 $19^{2} = 361 = 2 A 9_{11}$ .

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

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 $19^{2} = 361 = 2 A 9_{11}$ 。

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