Sieve of Eratosthenes

Star gazing at night—we sometimes wonder—How many stars are there in the universe? How many galaxies? How many natural numbers are there? All these are really not finite. They are infinite! Prime numbers are also infinite. A brilliant algorithm to find prime numbers was found by Eratosthenes of Cyrene, a Greek mathematician. Named after him, the Sieve of Eratosthenes algorithm can be very nicely expressed as follows:

scala> def numStream(n: Int): Stream[Int] = 
     | 
Stream.from(n)// 1
numStream: (n: Int)Stream[Int] 
scala> def sieve(stream: Stream[Int]): Stream[Int] = 
     |   stream.head #:: sieve((stream.tail) filter (x => x % stream.head != 0)) // 2
sieve: (stream: Stream[Int])Stream[Int] 
scala> val p = sieve(numStream(2)) 
p: Stream[Int] = Stream(2, ?) 
scala> (p take 5) foreach { println(_) } 
2 
3 
5 
7 
11 

By dissecting the code, we get the following findings:

1. We have a stream that generates successive numbers, starting off from the argument...