Generating Circular Primes

-- Project Euler: Problem 35  
--  
-- The number, 197, is called a circular prime because all rotations of the  
-- digits: 197, 971, and 719, are themselves prime.  
--  
-- There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37,  
-- 71, 73, 79, and 97.  
--  
-- How many circular primes are there below one million?  

import Data.Set (Set)  
import qualified Data.Set as Set  
import System (getArgs)  

-- How many circular primes do you want?  
limit = 1000000  

-- Memoized prime number generator.  
primes :: [Integer]  
primes = 2 : 3 : 5 : filter (not . hasFactor) [7..]  
  where hasFactor n = any (divides n) $ takeWhile (<= lowestFactor n) primes  
        divides n m = n `mod` m == 0  
        lowestFactor = ceiling . sqrt . fromIntegral  

-- Given an integer, return the number of digits in it.  
digits :: Integer -> Int  
digits n = ceiling $ logBase 10 $ fromIntegral (n + 1)  

-- Rotate number right:  
--   rotate 456 = 645  
--   rotate 3498 = 8349  
--  
-- Note that this does not work for numbers that have zeroes in them. But  
-- that's okay because any number with a zero in it is not a circular prime.  
-- Also, right rotation catches the zero before it disappears into the   
-- significant digit.  
rotate :: Integer -> Integer  
rotate n = strip_last_digit + (last_digit * tens_place)  
  where last_digit = n `mod` 10  
        strip_last_digit = (n - last_digit) `div` 10  
        tens_place = (10 ^ (digits n - 1))  

-- Set of million primes. This enables fast lookup in isCircularPrime.  
primeSet :: Set Integer  
primeSet = Set.fromList $ takeWhile (< limit) primes  

-- Returns True if number is prime.  
isPrime :: Integer -> Bool  
isPrime n = Set.member n primeSet  

-- Returns True if the number is a circular prime.  
isCircularPrime :: Integer -> Bool  
isCircularPrime n = all isPrime $ take (digits n) $ iterate rotate n  

-- Go.  
main :: IO ()  
main = print $ length $ filter isCircularPrime $ takeWhile (< limit) primes