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Problem 51


Problem 51


Prime digit replacements

By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.


素数数字替换

将两位数*3的第一个数字代换为任意数字,在九个可能值中有六个是素数:13、23、43、53、73和83。

将五位数56**3的第三和第四位数字代换为相同的任意数字,就得到了十个可能值中有七个是素数的最小例子,这个素数族是:56003、56113、56333、56443、56663、56773和56993。56003作为这一族中最小的成员,也是最小的满足这个性质的素数。

通过将部分数字(不一定相邻)代换为相同的任意数字,有时能够得到八个素数,求满足这一性质的最小素数。