Rearrange, add stuff

Author: tikipatel

Prime Numbers/README.md

@@ -1,10 +1,12 @@
 # Sieve Primes
 
+## Sieve of Eratosthenes
 Goal: To quickly get an array of prime numbers with a specified limit using the Sieve of Eratosthenes.
 
-The Sieve of Eratosthenes is an algorithm that finds all primes numbers up to any given limit.
+The Sieve of Eratosthenes is an algorithm that finds all primes numbers up to any given limit by iteratively marking composites of each prime starting with multiples of 2. [Seeing this algorithm][1] in action is the best way to understand it.
 
-Here's an implementation in Swift that should be easy to understand:
+
+### Implementation
 
 ```
 func primesTo(max: Int) -> [Int] {
@@ -30,10 +32,71 @@ print(primesArray)
 // prints [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
 ```
 
---*write some introduction to the Sieve of Eratosthenes*--
+## Sieve of Atkin
+
+Goal: To quickly get an array of prime numbers with a specified limit using the Sieve of Atkin.
+
+The Sieve of Atkin is an algorithm that finds all primes numbers up to any given limit.
+
+--*write some introduction to the Sieve of Atkin*--
+
+### Implementation
+
+```
+func primesTo(max: Int) -> [Int] {
+    var is_prime = [Bool](count: max + 1, repeatedValue: false)
+    is_prime[2] = true
+    is_prime[3] = true
+    let limit = Int(ceil(sqrt(Double(max))))
+    for x in 1...limit {
+        for y in 1...limit {
+            var num = 4 * x * x + y * y
+            if (num <= max && (num % 12 == 1 || num % 12 == 5)) {
+                is_prime[num] = true
+            }
+            num = 3 * x * x + y * y
+            if (num <= max && num % 12 == 7) {
+                is_prime[num] = true
+            }
+            if (x > y) {
+                num = 3 * x * x - y * y
+                if (num <= max && num % 12 == 11) {
+                    is_prime[num] = true
+                }
+            }
+        }
+    }
+    if limit > 5 {
+        for i in 5...limit {
+            if is_prime[i] {
+                for j in (i * i)..i..max {
+                    is_prime[j] = false
+                }
+            }
+        }
+    }
+    var primesArray = [Int]()
+    for (idx, val) in is_prime.enumerate() {
+        if val == true { primesArray.append(idx) }
+    }
+    return primesArray
+}
+```
+Put this code in a playground and test it like so:
+
+```
+let primesArray = primesTo(50)
+print(primesArray)
+// prints [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
+```
 
 ## See also
 
+
+[1]:https://en.wikipedia.org/wiki/File:Sieve_of_Eratosthenes_animation.gif#/media/File:Sieve_of_Eratosthenes_animation.gif
+
 [Sieve of Eratosthenes on Wikipedia](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes)
 
+[Sieve of Atkin on Wikipedia](https://en.wikipedia.org/wiki/Sieve_of_Atkin)
+
 *Written for Swift Algorithm Club by Pratikbhai Patel*