Add Sieve of Atkin algorithm for efficient prime generation (#12974)

* Add Sieve of Atkin algorithm for efficient prime generation

Implement the Sieve of Atkin algorithm as an alternative to the existing
Sieve of Eratosthenes. This modern algorithm offers better theoretical
complexity O(n / log log n) and uses quadratic forms for prime detection.

Features:
- Comprehensive docstring with algorithm explanation
- Type hints and input validation
- Extensive doctests covering edge cases
- Follows repository coding conventions

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* Enhance documentation for Sieve of Atkin

Added additional explanation about the Sieve of Atkin algorithm.

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

maths/sieve_of_atkin.py

@@ -0,0 +1,110 @@
+"""
+Sieve of Atkin algorithm for finding all prime numbers up to a given limit.
+
+The Sieve of Atkin is a modern variant of the ancient Sieve of Eratosthenes
+that is optimized for finding primes. It has better theoretical asymptotic
+complexity, especially for large ranges.  This is the basic, non-segmented
+form.
+
+Time Complexity: O(n / log log n)
+Space Complexity: O(n)
+
+Reference: https://en.wikipedia.org/wiki/Sieve_of_Atkin
+"""
+
+import math
+
+
+def sieve_of_atkin(limit: int) -> list[int]:
+    """
+    Generate all prime numbers up to a given limit using the Sieve of Atkin.
+
+    The Sieve of Atkin is an optimized version of the Sieve of Eratosthenes.
+    It uses a different set of quadratic forms to identify potential primes.
+
+    Args:
+        limit: Upper bound for finding primes (inclusive)
+
+    Returns:
+        List of prime numbers up to the given limit
+
+    Raises:
+        ValueError: If limit is negative
+
+    Examples:
+        >>> sieve_of_atkin(30)
+        [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
+        >>> sieve_of_atkin(10)
+        [2, 3, 5, 7]
+        >>> sieve_of_atkin(2)
+        [2]
+        >>> sieve_of_atkin(1)
+        []
+        >>> sieve_of_atkin(0)
+        []
+        >>> sieve_of_atkin(-5)
+        Traceback (most recent call last):
+        ...
+        ValueError: -5: Invalid input, please enter a non-negative integer.
+    """
+    if limit < 0:
+        msg = f"{limit}: Invalid input, please enter a non-negative integer."
+        raise ValueError(msg)
+
+    if limit < 2:
+        return []
+
+    # Initialize the sieve
+    sieve = [False] * (limit + 1)
+
+    # Mark 2 and 3 as prime if they're within the limit
+    if limit >= 2:
+        sieve[2] = True
+    if limit >= 3:
+        sieve[3] = True
+
+    # Main algorithm - mark numbers using quadratic forms
+    sqrt_limit = int(math.sqrt(limit)) + 1
+
+    for x in range(1, sqrt_limit):
+        for y in range(1, sqrt_limit):
+            # First quadratic form: 4x² + y²
+            n = 4 * x * x + y * y
+            if n <= limit and (n % 12 == 1 or n % 12 == 5):
+                sieve[n] = not sieve[n]
+
+            # Second quadratic form: 3x² + y²
+            n = 3 * x * x + y * y
+            if n <= limit and n % 12 == 7:
+                sieve[n] = not sieve[n]
+
+            # Third quadratic form: 3x² - y² (only when x > y)
+            if x > y:
+                n = 3 * x * x - y * y
+                if n <= limit and n % 12 == 11:
+                    sieve[n] = not sieve[n]
+
+    # Remove squares of primes
+    for r in range(5, sqrt_limit):
+        if sieve[r]:
+            square = r * r
+            for i in range(square, limit + 1, square):
+                sieve[i] = False
+
+    # Collect all primes
+    primes = []
+    for i in range(2, limit + 1):
+        if sieve[i]:
+            primes.append(i)
+
+    return primes
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Example usage
+    print("Prime numbers up to 30 using Sieve of Atkin:")
+    print(sieve_of_atkin(30))