## Spiral primes

### Problem 58

Published on Friday, 5th December 2003, 06:00 pm; Solved by 23980; Difficulty rating: 5%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%?