carmichael_number - add doctests (TheAlgorithms#10038)

* Replacing the generator with numpy vector operations from lu_decomposition.

* Revert "Replacing the generator with numpy vector operations from lu_decomposition."

This reverts commit ad217c6.

* Added doctests

* Update carmichael_number.py

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* Update carmichael_number.py

I make an empty commit to reset:
tests are failing.

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update carmichael_number.py

Made changes taking into account the addition: 
from maths.greatest_common_divisor import greatest_common_divisor.

Now instead of gcd it is used: greatest_common_divisor.

* Update carmichael_number.py

* Update carmichael_number.py

* Update carmichael_number.py

I added a check for 0 and negative numbers in the tests and the code itself. Simplified obtaining the final result.

* Update carmichael_number.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update maths/carmichael_number.py

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

* Update carmichael_number.py

* Update carmichael_number.py

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

Author: 3 people

maths/carmichael_number.py

@@ -10,10 +10,21 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
+
 from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
+    """
+
+    Examples:
+    >>> power(2, 15, 3)
+    2
+
+    >>> power(5, 1, 30)
+    5
+    """
+
     if y == 0:
         return 1
     temp = power(x, y // 2, mod) % mod
@@ -24,15 +35,47 @@ def power(x: int, y: int, mod: int) -> int:
 
 
 def is_carmichael_number(n: int) -> bool:
-    b = 2
-    while b < n:
-        if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
-            return False
-        b += 1
-    return True
+    """
+
+    Examples:
+    >>> is_carmichael_number(562)
+    False
+
+    >>> is_carmichael_number(561)
+    True
+
+    >>> is_carmichael_number(5.1)
+    Traceback (most recent call last):
+         ...
+    ValueError: Number 5.1 must instead be a positive integer
+
+    >>> is_carmichael_number(-7)
+    Traceback (most recent call last):
+         ...
+    ValueError: Number -7 must instead be a positive integer
+
+    >>> is_carmichael_number(0)
+    Traceback (most recent call last):
+         ...
+    ValueError: Number 0 must instead be a positive integer
+    """
+
+    if n <= 0 or not isinstance(n, int):
+        msg = f"Number {n} must instead be a positive integer"
+        raise ValueError(msg)
+
+    return all(
+        power(b, n - 1, n) == 1
+        for b in range(2, n)
+        if greatest_common_divisor(b, n) == 1
+    )
 
 
 if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
     number = int(input("Enter number: ").strip())
     if is_carmichael_number(number):
         print(f"{number} is a Carmichael Number.")