feat(maths): add Sieve of Atkin prime sieve implementation

Author: idrisibrahimerten

maths/sieve_of_atkin.py

@@ -0,0 +1,45 @@
+from typing import List
+
+
+def sieve_of_atkin(limit: int) -> List[int]:
+    if not isinstance(limit, int) or limit < 2:
+        raise ValueError("limit must be an integer >= 2")
+
+    # Initialize the sieve array
+    sieve = [False] * (limit + 1)
+    results: List[int] = []
+
+    # Preliminary marking based on quadratic forms
+    from math import sqrt
+
+    sqrt_limit = int(sqrt(limit)) + 1
+    for x in range(1, sqrt_limit):
+        for y in range(1, sqrt_limit):
+            n = 4 * x * x + y * y
+            if n <= limit and n % 12 in (1, 5):
+                sieve[n] = not sieve[n]
+            n = 3 * x * x + y * y
+            if n <= limit and n % 12 == 7:
+                sieve[n] = not sieve[n]
+            n = 3 * x * x - y * y
+            if x > y and n <= limit and n % 12 == 11:
+                sieve[n] = not sieve[n]
+
+    # Mark all multiples of squares as non-prime
+    for n in range(5, sqrt_limit):
+        if sieve[n]:
+            step = n * n
+            for k in range(step, limit + 1, step):
+                sieve[k] = False
+
+    # Compile the list of primes
+    if limit >= 2:
+        results.extend([2, 3])
+    results.extend([i for i in range(5, limit + 1) if sieve[i]])
+    return results
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()
+    print("All doctests passed!")