# [Project Euler] – Problem 58

May 21, 2011
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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%?

?View Code RSPLUS

 ```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) }```

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