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Problem 58

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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 $8/13\approx 62\mathrm{\%}$.

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 $10\mathrm{\%}$?

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螺旋素数

从$1$开始按照逆时针方向摆放自然数，可以构造出如下边长为$7$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

有趣的是，所有的奇数平方都在这个螺旋方阵的右下对角线上，更有趣的是，在两条对角线上的$13$个数中一共有$8$个素数，比例达到$8/13\approx 62\mathrm{\%}$。

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

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