Added Germain primes algorithm to the maths folder (#10120)

* Added algorithm for Germain Primes to maths folder

* Fixed test errors Germain primes.

* Formatting Germain primes after pre-commit

* Fixed path to maths

* Update maths/germain_primes.py

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

* Update maths/germain_primes.py

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

* Added function for safe primes

* Update maths/germain_primes.py

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

* Apply suggestions from code review

---------

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

Author: 3 people

Diff for: maths/germain_primes.py

@@ -0,0 +1,72 @@
+"""
+A Sophie Germain prime is any prime p, where 2p + 1 is also prime.
+The second number, 2p + 1 is called a safe prime.
+
+Examples of Germain primes include: 2, 3, 5, 11, 23
+
+Their corresponding safe primes: 5, 7, 11, 23, 47
+https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes
+"""
+
+from maths.prime_check import is_prime
+
+
+def is_germain_prime(number: int) -> bool:
+    """Checks if input number and 2*number + 1 are prime.
+
+    >>> is_germain_prime(3)
+    True
+    >>> is_germain_prime(11)
+    True
+    >>> is_germain_prime(4)
+    False
+    >>> is_germain_prime(23)
+    True
+    >>> is_germain_prime(13)
+    False
+    >>> is_germain_prime(20)
+    False
+    >>> is_germain_prime('abc')
+    Traceback (most recent call last):
+        ...
+    TypeError: Input value must be a positive integer. Input value: abc
+    """
+    if not isinstance(number, int) or number < 1:
+        msg = f"Input value must be a positive integer. Input value: {number}"
+        raise TypeError(msg)
+
+    return is_prime(number) and is_prime(2 * number + 1)
+
+
+def is_safe_prime(number: int) -> bool:
+    """Checks if input number and (number - 1)/2 are prime.
+    The smallest safe prime is 5, with the Germain prime is 2.
+
+    >>> is_safe_prime(5)
+    True
+    >>> is_safe_prime(11)
+    True
+    >>> is_safe_prime(1)
+    False
+    >>> is_safe_prime(2)
+    False
+    >>> is_safe_prime(3)
+    False
+    >>> is_safe_prime(47)
+    True
+    >>> is_safe_prime('abc')
+    Traceback (most recent call last):
+        ...
+    TypeError: Input value must be a positive integer. Input value: abc
+    """
+    if not isinstance(number, int) or number < 1:
+        msg = f"Input value must be a positive integer. Input value: {number}"
+        raise TypeError(msg)
+
+    return (number - 1) % 2 == 0 and is_prime(number) and is_prime((number - 1) // 2)
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()