From 0af87f8afec29e461e05ce66975436c26fbc63d4 Mon Sep 17 00:00:00 2001
From: Manpreet Singh <63737630+ManpreetSingh2004@users.noreply.github.com>
Date: Wed, 18 Oct 2023 17:30:09 +0530
Subject: [PATCH 1/4] Fix integer overflow on windows

---
 project_euler/problem_072/sol1.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/project_euler/problem_072/sol1.py b/project_euler/problem_072/sol1.py
index 5a28be564556..9390ce9f658c 100644
--- a/project_euler/problem_072/sol1.py
+++ b/project_euler/problem_072/sol1.py
@@ -36,7 +36,7 @@ def solution(limit: int = 1_000_000) -> int:
     """
 
     # generating an array from -1 to limit
-    phi = np.arange(-1, limit)
+    phi = np.arange(-1, limit, dtype=np.int64)
 
     for i in range(2, limit + 1):
         if phi[i] == i - 1:

From 89bdb64f15442501277d282532c49afe66e6c614 Mon Sep 17 00:00:00 2001
From: Manpreet Singh <63737630+ManpreetSingh2004@users.noreply.github.com>
Date: Wed, 18 Oct 2023 17:49:32 +0530
Subject: [PATCH 2/4] Use slicing for element selection

---
 project_euler/problem_072/sol1.py | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/project_euler/problem_072/sol1.py b/project_euler/problem_072/sol1.py
index 9390ce9f658c..ed079efe7d05 100644
--- a/project_euler/problem_072/sol1.py
+++ b/project_euler/problem_072/sol1.py
@@ -18,7 +18,7 @@
 function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000
 Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve.
 
-Time: 1 sec
+Time: 0.4 sec
 """
 
 import numpy as np
@@ -40,8 +40,7 @@ def solution(limit: int = 1_000_000) -> int:
 
     for i in range(2, limit + 1):
         if phi[i] == i - 1:
-            ind = np.arange(2 * i, limit + 1, i)  # indexes for selection
-            phi[ind] -= phi[ind] // i
+            phi[2 * i :: i] -= phi[2 * i :: i] // i
 
     return np.sum(phi[2 : limit + 1])
 

From 0e3ad313672c6d61a84b41bdd99f4ab04a5985ca Mon Sep 17 00:00:00 2001
From: Manpreet Singh <63737630+ManpreetSingh2004@users.noreply.github.com>
Date: Wed, 18 Oct 2023 17:56:15 +0530
Subject: [PATCH 3/4] Add runtime in sol2.py

---
 project_euler/problem_072/sol2.py | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/project_euler/problem_072/sol2.py b/project_euler/problem_072/sol2.py
index 08e92c18bb3a..631d68e31130 100644
--- a/project_euler/problem_072/sol2.py
+++ b/project_euler/problem_072/sol2.py
@@ -20,6 +20,9 @@
 def solution(limit: int = 1000000) -> int:
     """
     Return the number of reduced proper fractions with denominator less than limit.
+
+    Time: 1 sec
+
     >>> solution(8)
     21
     >>> solution(1000)

From a4ab0c7756d5c8d1ed9293bde6f0f75be43637be Mon Sep 17 00:00:00 2001
From: Manpreet Singh <63737630+ManpreetSingh2004@users.noreply.github.com>
Date: Wed, 18 Oct 2023 18:06:24 +0530
Subject: [PATCH 4/4] Separate out prime number generator from solution2

---
 project_euler/problem_072/sol2.py | 31 +++++++++++++++++++++++++------
 1 file changed, 25 insertions(+), 6 deletions(-)

diff --git a/project_euler/problem_072/sol2.py b/project_euler/problem_072/sol2.py
index 631d68e31130..2b4fb6b994da 100644
--- a/project_euler/problem_072/sol2.py
+++ b/project_euler/problem_072/sol2.py
@@ -17,6 +17,30 @@
 """
 
 
+def set_generate_primes(limit: int) -> set:
+    """
+    Returns prime numbers below max_number.
+
+    >>> set_generate_primes(10)
+    {2, 3, 5, 7}
+    >>> set_generate_primes(2)
+    set()
+    """
+
+    if limit <= 2:
+        return set()
+
+    primes = set(range(3, limit, 2))
+    primes.add(2)
+
+    for p in range(3, limit, 2):
+        if p not in primes:
+            continue
+        primes.difference_update(set(range(p * p, limit, p)))
+
+    return primes
+
+
 def solution(limit: int = 1000000) -> int:
     """
     Return the number of reduced proper fractions with denominator less than limit.
@@ -28,13 +52,8 @@ def solution(limit: int = 1000000) -> int:
     >>> solution(1000)
     304191
     """
-    primes = set(range(3, limit, 2))
-    primes.add(2)
-    for p in range(3, limit, 2):
-        if p not in primes:
-            continue
-        primes.difference_update(set(range(p * p, limit, p)))
 
+    primes = set_generate_primes(limit)
     phi = [float(n) for n in range(limit + 1)]
 
     for p in primes: