From 16d092aec56a4cf636aced7fe3b3028c44a9d48c Mon Sep 17 00:00:00 2001
From: Thabhelo <50872400+Thabhelo@users.noreply.github.com>
Date: Wed, 21 Aug 2024 22:42:38 -0500
Subject: [PATCH] Create sieve_of_eratosthenes.py

Added a new implementation of the 'Sieve of Eratosthenes' algorithm to the Maths subdirectory. This efficient algorithm finds all prime numbers up to a given limit.
---
 Maths/sieve_of_eratosthenes.py | 50 ++++++++++++++++++++++++++++++++++
 1 file changed, 50 insertions(+)
 create mode 100644 Maths/sieve_of_eratosthenes.py

diff --git a/Maths/sieve_of_eratosthenes.py b/Maths/sieve_of_eratosthenes.py
new file mode 100644
index 0000000..0ff281b
--- /dev/null
+++ b/Maths/sieve_of_eratosthenes.py
@@ -0,0 +1,50 @@
+'''There is a classic algorithm in number theory that I thought can be a great addition to this Maths subdirectory: The 'Sieve of Eratosthenes'. 
+It's an efficient way to find all prime numbers up to a given limit.
+'''
+
+def sieve_of_eratosthenes(n):
+  """
+  Finds all prime numbers up to a given number n using the Sieve of Eratosthenes.
+
+  This algorithm efficiently determines all primes less than or equal to n by iteratively marking multiples of primes as non-prime.
+
+  Args:
+    n: The upper limit for finding prime numbers (inclusive).
+
+  Returns:
+    A list containing all the prime numbers less than or equal to n.
+  """
+
+  # Create a list of booleans, where primes[i] is True if i is prime, False otherwise.
+  primes = [True] * (n + 1)
+
+  # 0 and 1 are not prime by definition
+  primes[0] = primes[1] = False
+
+  # Efficiently iterate only up to the square root of n
+  # Numbers greater than the square root of n cannot have prime factors
+  # greater than the square root
+  for p in range(2, int(n**0.5) + 1):
+    if primes[p]:  # If p is prime
+      # Mark all multiples of p as non-prime (composite)
+      for i in range(p * p, n + 1, p):
+        primes[i] = False
+
+  # Return a list of all remaining primes (where primes[i] is True)
+  return [i for i in range(2, n + 1) if primes[i]]
+
+# This is how you would use it, say:
+n = 100
+prime_numbers = sieve_of_eratosthenes(n)
+print(prime_numbers)
+
+'''Complexity Analysis:
+
+For the Time Complexity:
+
+Best Case: O(n) (when n is small)
+Average Case: O(n log log n)
+Worst Case: O(n log log n) 👍 ✅ 
+
+For Space Complexity: O(n)
+'''