[Project Euler] – Problem 58

May 21, 2011

(This article was first published on YGC, and kindly contributed to R-bloggers)

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

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)
}