## Question

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 $\frac{8}{13} \approx 62\%$.

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%?

isPrime :: Int -> Bool
isPrime n | n < 1 = False
| otherwise = not $or [n rem x == 0 | x <- [2..floor$ sqrt $fromIntegral n]] primeCorners :: Int -> Int primeCorners n = sum [1 | x <- [1..3], isPrime$ n^2 - x*(n - 1)]

expansion :: [(Int, Int)]
expansion = scanl (\(p, t) x -> (p + primeCorners x, t + 4)) (0, 1) [3,5..]

main :: IO ()
main = print $3 + 2 * (length$ takeWhile id $tail [10*p >= t | (p, t) <- expansion]) $ ghc -O2 -o spiral-primes spiral-primes.hs
$time ./spiral-primes real 0m1.372s user 0m1.360s sys 0m0.000s ## Python #!/usr/bin/env python import math from collections import defaultdict from functools import reduce def factorize(n): if n < 1: raise ValueError('fact() argument should be >= 1') if n == 1: return [] # special case res = [] # iterate over all even numbers first. while n % 2 == 0: res.append(2) n //= 2 # try odd numbers up to sqrt(n) limit = math.sqrt(n+1) i = 3 while i <= limit: if n % i == 0: res.append(i) n //= i limit = math.sqrt(n+i) else: i += 2 if n != 1: res.append(n) return res def num_divisors(n): factors = sorted(factorize(n)) histogram = defaultdict(int) for factor in factors: histogram[factor] += 1 # number of divisors is equal to product of # incremented exponents of prime factors from operator import mul try: return reduce(mul, [exponent + 1 for exponent in list(histogram.values())]) except: return 1 def is_prime(num): if num % 2 == 0: return False if num % 3 == 0 and num != 3: return False if num_divisors(num) == 2 and num > 1: return True else: return False def spiral_diagonals(): n = 1 step = 2 since_last = 0 while True: yield n n += step since_last += 1 if since_last == 4: step += 2 since_last = 0 def main(): level = 0 primes = 0 for i, n in enumerate(spiral_diagonals()): if (i-1) % 4 == 0: level += 1 if is_prime(n): primes += 1 side_length = (2 * level) + 1 ratio = float(primes) / float(i+1) if 0 < ratio < 0.1: print(side_length) return if __name__ == "__main__": main() $ time python3 spiral-primes.py
real   0m15.097s
user   0m14.996s
sys    0m0.012s

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