i have used sieve of Eratosthenes Update prime_numbers.py

basically let assum we have [1...6] so it is obvious that if 6 isdiv by 6 then it can't be a prime number which will result less time complexcity

Author: Nerdgirl50

maths/prime_numbers.py

@@ -1,63 +1,5 @@
 import math
 from collections.abc import Generator
-
-
-def slow_primes(max_n: int) -> Generator[int, None, None]:
-    """
-    Return a list of all primes numbers up to max.
-    >>> list(slow_primes(0))
-    []
-    >>> list(slow_primes(-1))
-    []
-    >>> list(slow_primes(-10))
-    []
-    >>> list(slow_primes(25))
-    [2, 3, 5, 7, 11, 13, 17, 19, 23]
-    >>> list(slow_primes(11))
-    [2, 3, 5, 7, 11]
-    >>> list(slow_primes(33))
-    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
-    >>> list(slow_primes(1000))[-1]
-    997
-    """
-    numbers: Generator = (i for i in range(1, (max_n + 1)))
-    for i in (n for n in numbers if n > 1):
-        for j in range(2, i):
-            if (i % j) == 0:
-                break
-        else:
-            yield i
-
-
-def primes(max_n: int) -> Generator[int, None, None]:
-    """
-    Return a list of all primes numbers up to max.
-    >>> list(primes(0))
-    []
-    >>> list(primes(-1))
-    []
-    >>> list(primes(-10))
-    []
-    >>> list(primes(25))
-    [2, 3, 5, 7, 11, 13, 17, 19, 23]
-    >>> list(primes(11))
-    [2, 3, 5, 7, 11]
-    >>> list(primes(33))
-    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
-    >>> list(primes(1000))[-1]
-    997
-    """
-    numbers: Generator = (i for i in range(1, (max_n + 1)))
-    for i in (n for n in numbers if n > 1):
-        # only need to check for factors up to sqrt(i)
-        bound = int(math.sqrt(i)) + 1
-        for j in range(2, bound):
-            if (i % j) == 0:
-                break
-        else:
-            yield i
-
-
 def fast_primes(max_n: int) -> Generator[int, None, None]:
     """
     Return a list of all primes numbers up to max.
@@ -76,19 +18,14 @@ def fast_primes(max_n: int) -> Generator[int, None, None]:
     >>> list(fast_primes(1000))[-1]
     997
     """
-    numbers: Generator = (i for i in range(1, (max_n + 1), 2))
-    # It's useless to test even numbers as they will not be prime
-    if max_n > 2:
-        yield 2  # Because 2 will not be tested, it's necessary to yield it now
-    for i in (n for n in numbers if n > 1):
-        bound = int(math.sqrt(i)) + 1
-        for j in range(3, bound, 2):
-            # As we removed the even numbers, we don't need them now
-            if (i % j) == 0:
-                break
-        else:
-            yield i
-
+    #it is useless to calculate who is already 2 multiple 
+   def countPrimes(self, max_n: int) -> int:
+        seen, ans = [0] * max_n, 0
+        for num in range(2, max_n):
+            if seen[num]: continue
+            ans += 1
+            seen[num*num:n:num] = [1] * ((max_n - 1) // num - num + 1)
+        return ans
 
 def benchmark():
     """