Problem 58
Spiral primes
Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.
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 $8/13 \approx 62\%$.
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。
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$8/13 \approx 62\%$。
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$10\%$时，螺旋数阵的边长是多少？