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Problem 58


Problem 58


Spiral primes

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

**37** 36 35 34 33 32 **31** 38 **17** 16 15 14 **13** 30 39 18  **5**  4  **3** 12 29 40 19  6  1  2 11 28 41 20  **7**  8  9 10 27 42 21 22 23 24 25 26 **43** 44 45 46 47 48 49

It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.

If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?


螺旋素数
从1开始逆时针螺旋着摆放自然数,我们可以构造出一个边长为7的螺旋数阵。

**37** 36 35 34 33 32 **31** 38 **17** 16 15 14 **13** 30 39 18  **5**  4  **3** 12 29 40 19  6  1  2 11 28 41 20  **7**  8  9 10 27 42 21 22 23 24 25 26 **43** 44 45 46 47 48 49

可以发现,所有的奇数平方都在这个螺旋方针的右下对角线上,更有趣的是,在所有对角线上一共有8个素数,比例达到8/13 ≈ 62%。

在这个方阵外面完整地再加上一层,就能构造出一个边长为9的螺旋方阵。如果不断重复这个过程,当对角线上素数的比例第一次低于10%时,螺旋数阵的边长是多少?