#58 - Spiral primes

in Project Euler on 5%

Starting with 1 and spiral 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%?

We first need to construct the spiral of numbers. The side length can only be odd, and furthermore, to get from one corner number to the other, you add one less than the length of the side you are on. For example, from 13 to 17, you need to add 4, while the side length is 5.

Our methodology is to keep generating primes from one square to the next, calculate the 4 corner numbers, and see how many are primes. We keep track of the total number of diagonal numbers and the total number of primes.

# file: "problem058.py"
# Side lengths are always
# odd. Keep track of
# primes on diagonals...
sideLen = 7
primesOnDiags = 8
totalDiags = 13
while primesOnDiags / totalDiags >= 0.1:
    # Put one complete layer.
    diagonalNums = [sideLen ** 2 + (sideLen + 1) * i for i in range(1, 5)]
    # Generate primes in this layer...
    primesInLayer = primesieve.primes(sideLen ** 2, (sideLen + 2) ** 2)
    # Go through each diagonal number, and see if it's
    # in the prime set...increment the primesOnDiags if it is...
    primesOnDiags += sum(diag in primesInLayer for diag in diagonalNums)
    # Number of diagonal numbers increases by 4
    totalDiags += 4
    # Side length increases by 2...
    sideLen += 2

print(sideLen)

The output is,

26241
3.9868424743338373 seconds.

Therefore, a side length of 26241 is needed before the ratio falls below 10%.