Problem 58 - Investigate the number of primes that lie on the diagonals of the spiral grid.

링크

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%?

python

def isprime(n): if n == 1: return 0 if n < 4: return 1 if (n & 1) == 0: return 0 if n < 9: return 1 if n % 3 == 0: return 0 r = int(n**0.5) f = 5 while f <= r: if (n % f) == 0 or (n % (f + 2)) == 0: return 0 f += 6 return 1 len = 3 cnt = 3 while cnt / (2.0*len-1) > 0.1: len += 2 cnt += isprime(len*len) cnt += isprime(len*len - (len-1)) cnt += isprime(len*len - 2*(len-1)) cnt += isprime(len*len - 3*(len-1)) print len