[Project Euler] – Problem 58
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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%?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | side.length = 1 x <- 1 iter <- 1 ratio = 0.6 isp.n <- 0 N <- 0 while(ratio > 0.1) { last.x <- x[length(x)] side.length <- side.length + 2 x <- rep(last.x,4 ) + c(2,4,6,8)* iter iter <- iter + 1 isp <- gmp::isprime(x) isp.n <- isp.n + sum(as.logical(isp)) N <- N + 4 ratio <- isp.n/N print(side.length) } |
Answer: 26241
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