carmichael_number - add doctests (#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 ad217c66165898d62b76cc89ba09c2d7049b6448.

* Added doctests

* Update carmichael_number.py

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

for more information, see https://pre-commit.ci

* Update carmichael_number.py

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

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

for more information, see https://pre-commit.ci

* 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

for more information, see https://pre-commit.ci

* Update maths/carmichael_number.py

Co-authored-by: Tianyi Zheng <[email protected]>

* 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 <[email protected]>