Update carmichael.py

Some performance improvements.

Author: gunslingster

Carmichael Numbers/carmichael.py

@@ -1,31 +1,51 @@
 import math
 
-def isPrime(n):
-    i = 2
-    while i <= math.sqrt(n):
-        if n % i == 0:
+def isPrime(num):
+    if num <= 3:
+        return num > 1
+    if num % 2 == 0:
+        return False
+    if num % 3 == 0:
+        return False
+    i = 5
+    while i <= math.sqrt(num):
+        if num % (i) == 0 or num % (i+2) == 0:
             return False
-        i += 1
+        i += 6
     return True
-    
+
+def isCoprime(a, b):
+    return math.gcd(a, b) == 1
+
 def isCarmichael(n):
     if isPrime(n):
         return False
-    for i in range(1, n):
-        if i**n  % n == i:
-            continue
-        else:
-            return False
+    if n % 2 == 0:
+        for i in range(1, n, 2):
+            if isCoprime(n, i):
+                if i**(n-1) % n == 1:
+                    continue
+                else:
+                    return False
+    else:
+        for i in range(1, n):
+            if isCoprime(n, i):
+                if i**(n-1) % n == 1:
+                    continue
+                else:
+                    return False
     return True
 
 def main():
     n = int(input('Enter n: '))
     count = -1
-    curr = 0
+    curr = 3
     while count < n:
-        curr += 1
+        curr += 2
         if isCarmichael(curr):
             count += 1
     print(curr)
 
-main()
+if __name__ == '__main__':
+    main()
+