


from math import sqrt




def     UpperCamelCase    (      _a		)					->							bool:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (
         number >= 0
     ), "'number' must been an int and positive"

     lowercase_       :Dict     =       True

     # 0 and 1 are none primes.
     if number <= 1:
          lowercase_       :Union[str, Any]     =       False

     for divisor in range(2       ,		int(round(sqrt(_a		)		)		) + 1		):
          # if 'number' divisible by 'divisor' then sets 'status'
          # of false and break up the loop.
          if number % divisor == 0:
               lowercase_       :Tuple     =       False
               break

    # precondition
     assert isinstance(_a       ,		_a		), "'status' must been from type bool"

     return status




def     UpperCamelCase    (      _a		)					->							List[str]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n > 2), "'N' must been an int and > 2"

     # beginList: contains all natural numbers from 2 up to N
     lowercase_       :int     =       list(range(2       ,		n + 1		)		)

     lowercase_       :Tuple     =       []  # this list will be returns.

     # actual sieve of erathostenes
     for i in range(len(_a		)		):
          for j in range(i + 1       ,		len(_a		)		):
               if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
                    lowercase_       :List[str]     =       0

    # filters actual prime numbers.
     lowercase_       :Optional[int]     =       [x for x in begin_list if x != 0]

     # precondition
     assert isinstance(_a       ,		_a		), "'ans' must been from type list"

     return ans




def     UpperCamelCase    (      _a		)					->							Optional[int]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n > 2), "'N' must been an int and > 2"

     lowercase_       :Any     =       []

     # iterates over all numbers between 2 up to N+1
     # if a number is prime then appends to list 'ans'
     for number in range(2       ,		n + 1		):
          if is_prime(_a		):
               ans.append(_a		)

    # precondition
     assert isinstance(_a       ,		_a		), "'ans' must been from type list"

     return ans




def     UpperCamelCase    (      _a		)					->							List[Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and number >= 0, "'number' must been an int and >= 0"

     lowercase_       :int     =       []  # this list will be returns of the function.

     # potential prime number factors.

     lowercase_       :str     =       2

     lowercase_       :Optional[Any]     =       number

     if number == 0 or number == 1:
          ans.append(_a		)

     # if 'number' not prime then builds the prime factorization of 'number'
     elif not is_prime(_a		):
          while quotient != 1:
               if is_prime(_a		) and (quotient % factor == 0):
                    ans.append(_a		)
                    quotient /= factor
               else:
                    factor += 1

     else:
          ans.append(_a		)

     # precondition
     assert isinstance(_a       ,		_a		), "'ans' must been from type list"

     return ans




def     UpperCamelCase    (      _a		)					->							List[str]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (
         number >= 0
     ), "'number' bust been an int and >= 0"

     lowercase_       :Dict     =       0

     # prime factorization of 'number'
     lowercase_       :Tuple     =       prime_factorization(_a		)

     lowercase_       :str     =       max(_a		)

     # precondition
     assert isinstance(_a       ,		_a		), "'ans' must been from type int"

     return ans




def     UpperCamelCase    (      _a		)					->							Optional[Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (
         number >= 0
     ), "'number' bust been an int and >= 0"

     lowercase_       :List[str]     =       0

     # prime factorization of 'number'
     lowercase_       :Tuple     =       prime_factorization(_a		)

     lowercase_       :Optional[int]     =       min(_a		)

     # precondition
     assert isinstance(_a       ,		_a		), "'ans' must been from type int"

     return ans




def     UpperCamelCase    (      _a		)					->							int:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		), "'number' must been an int"
     assert isinstance(number % 2 == 0       ,		_a		), "compare bust been from type bool"

     return number % 2 == 0




def     UpperCamelCase    (      _a		)					->							List[Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		), "'number' must been an int"
     assert isinstance(number % 2 != 0       ,		_a		), "compare bust been from type bool"

     return number % 2 != 0




def     UpperCamelCase    (      _a		)					->							List[Any]:
     '''simple docstring'''





     assert (
         isinstance(_a       ,		_a		) and (number > 2) and is_even(_a		)
     ), "'number' must been an int, even and > 2"

     lowercase_       :str     =       []  # this list will returned

     # creates a list of prime numbers between 2 up to 'number'
     lowercase_       :Union[str, Any]     =       get_prime_numbers(_a		)
     lowercase_       :List[Any]     =       len(_a		)

     # run variable for while-loops.
     lowercase_       :List[Any]     =       0
     lowercase_       :Optional[int]     =       None

     # exit variable. for break up the loops
     lowercase_       :List[str]     =       True

     while i < len_pn and loop:
          lowercase_       :Tuple     =       i + 1

          while j < len_pn and loop:
               if prime_numbers[i] + prime_numbers[j] == number:
                    lowercase_       :Optional[Any]     =       False
                    ans.append(prime_numbers[i]		)
                    ans.append(prime_numbers[j]		)

               j += 1

          i += 1

     # precondition
     assert (
         isinstance(_a       ,		_a		)
         and (len(_a		) == 2)
         and (ans[0] + ans[1] == number)
         and is_prime(ans[0]		)
         and is_prime(ans[1]		)
     ), "'ans' must contains two primes. And sum of elements must been eq 'number'"

     return ans




def     UpperCamelCase    (      _a       ,		_a		)					->							Tuple:
     '''simple docstring'''





     assert (
         isinstance(_a       ,		_a		)
         and isinstance(_a       ,		_a		)
         and (numbera >= 0)
         and (numbera >= 0)
     ), "'number1' and 'number2' must been positive integer."

     lowercase_       :Dict     =       0

     while numbera != 0:
          lowercase_       :int     =       numbera % numbera
          lowercase_       :Optional[int]     =       numbera
          lowercase_       :Optional[Any]     =       rest

     # precondition
     assert isinstance(_a       ,		_a		) and (
         numbera >= 0
     ), "'number' must been from type int and positive"

     return numbera




def     UpperCamelCase    (      _a       ,		_a		)					->							int:
     '''simple docstring'''





     assert (
         isinstance(_a       ,		_a		)
         and isinstance(_a       ,		_a		)
         and (numbera >= 1)
         and (numbera >= 1)
     ), "'number1' and 'number2' must been positive integer."

     lowercase_       :List[str]     =       1  # actual answer that will be return.

     # for kgV (x,1)
     if numbera > 1 and numbera > 1:
          # builds the prime factorization of 'number1' and 'number2'
          lowercase_       :Tuple     =       prime_factorization(_a		)
          lowercase_       :Optional[Any]     =       prime_factorization(_a		)

     elif numbera == 1 or numbera == 1:
          lowercase_       :int     =       []
          lowercase_       :int     =       []
          lowercase_       :Union[str, Any]     =       max(_a       ,		_a		)

     lowercase_       :List[str]     =       0
     lowercase_       :str     =       0

     lowercase_       :Optional[Any]     =       []  # captured numbers int both 'primeFac1' and 'primeFac2'

     # iterates through primeFac1
     for n in prime_fac_a:
          if n not in done:
               if n in prime_fac_a:
                    lowercase_       :str     =       prime_fac_a.count(_a		)
                    lowercase_       :Optional[int]     =       prime_fac_a.count(_a		)

                    for _ in range(max(_a       ,		_a		)		):
                         ans *= n

               else:
                    lowercase_       :Optional[int]     =       prime_fac_a.count(_a		)

                    for _ in range(_a		):
                         ans *= n

               done.append(_a		)

    # iterates through primeFac2
     for n in prime_fac_a:
          if n not in done:
               lowercase_       :Union[str, Any]     =       prime_fac_a.count(_a		)

               for _ in range(_a		):
                    ans *= n

               done.append(_a		)

    # precondition
     assert isinstance(_a       ,		_a		) and (
         ans >= 0
     ), "'ans' must been from type int and positive"

     return ans




def     UpperCamelCase    (      _a		)					->							Union[str, Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n >= 0), "'number' must been a positive int"

     lowercase_       :Tuple     =       0
     lowercase_       :Union[str, Any]     =       2  # this variable holds the answer

     while index < n:
          index += 1

          ans += 1  # counts to the next number

          # if ans not prime then
          # runs to the next prime number.
          while not is_prime(_a		):
               ans += 1

    # precondition
     assert isinstance(_a       ,		_a		) and is_prime(
         _a		), "'ans' must been a prime number and from type int"

     return ans




def     UpperCamelCase    (      _a       ,		_a		)					->							Optional[Any]:
     '''simple docstring'''





     assert (
         is_prime(_a		) and is_prime(_a		) and (p_number_a < p_number_a)
     ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

     lowercase_       :Tuple     =       p_number_a + 1  # jump to the next number

     lowercase_       :List[str]     =       []  # this list will be returns.

     # if number is not prime then
     # fetch the next prime number.
     while not is_prime(_a		):
          number += 1

     while number < p_number_a:
          ans.append(_a		)

          number += 1

          # fetch the next prime number.
          while not is_prime(_a		):
               number += 1

    # precondition
     assert (
         isinstance(_a       ,		_a		)
         and ans[0] != p_number_a
         and ans[len(_a		) - 1] != p_number_a
     ), "'ans' must been a list without the arguments"

     # 'ans' contains not 'pNumber1' and 'pNumber2' !
     return ans




def     UpperCamelCase    (      _a		)					->							Union[str, Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n >= 1), "'n' must been int and >= 1"

     lowercase_       :Tuple     =       []  # will be returned.

     for divisor in range(1       ,		n + 1		):
          if n % divisor == 0:
               ans.append(_a		)

    # precondition
     assert ans[0] == 1 and ans[len(_a		) - 1] == n, "Error in function getDivisiors(...)"

     return ans




def     UpperCamelCase    (      _a		)					->							List[Any]:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (
         number > 1
     ), "'number' must been an int and >= 1"

     lowercase_       :int     =       get_divisors(_a		)

     # precondition
     assert (
         isinstance(_a       ,		_a		)
         and (divisors[0] == 1)
         and (divisors[len(_a		) - 1] == number)
     ), "Error in help-function getDivisiors(...)"

     # summed all divisors up to 'number' (exclusive), hence [:-1]
     return sum(divisors[:-1]		) == number




def     UpperCamelCase    (      _a       ,		_a		)					->							Tuple:
     '''simple docstring'''





     assert (
         isinstance(_a       ,		_a		)
         and isinstance(_a       ,		_a		)
         and (denominator != 0)
     ), "The arguments must been from type int and 'denominator' != 0"

     # build the greatest common divisor of numerator and denominator.
     lowercase_       :Union[str, Any]     =       gcd(abs(_a		)       ,		abs(_a		)		)

     # precondition
     assert (
         isinstance(_a       ,		_a		)
         and (numerator % gcd_of_fraction == 0)
         and (denominator % gcd_of_fraction == 0)
     ), "Error in function gcd(...,...)"

     return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)




def     UpperCamelCase    (      _a		)					->							Tuple:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n >= 0), "'n' must been a int and >= 0"

     lowercase_       :str     =       1  # this will be return.

     for factor in range(1       ,		n + 1		):
          ans *= factor

     return ans




def     UpperCamelCase    (      _a		)					->							str:
     '''simple docstring'''





     assert isinstance(_a       ,		_a		) and (n >= 0), "'n' must been an int and >= 0"

     lowercase_       :str     =       0
     lowercase_       :Optional[int]     =       1
     lowercase_       :str     =       1  # this will be return

     for _ in range(n - 1		):
          lowercase_       :int     =       ans
          ans += fiba
          lowercase_       :List[Any]     =       tmp

     return ans





from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
SCREAMING_SNAKE_CASE       :		int         =				299_792_458

# Symbols
SCREAMING_SNAKE_CASE     ,		SCREAMING_SNAKE_CASE     ,		SCREAMING_SNAKE_CASE     ,		SCREAMING_SNAKE_CASE       :		Optional[Any]         =				symbols("ct x y z")




def     UpperCamelCase    (      _a		)					->							float:
     '''simple docstring'''





     if velocity > c:
          raise ValueError('''Speed must not exceed light speed 299,792,458 [m/s]!'''		)
     elif velocity < 1:
          # Usually the speed should be much higher than 1 (c order of magnitude)
          raise ValueError('''Speed must be greater than or equal to 1!'''		)

     return velocity / c




def     UpperCamelCase    (      _a		)					->							float:
     '''simple docstring'''





     return 1 / sqrt(1 - beta(_a		) ** 2		)




def     UpperCamelCase    (      _a		)					->							np.ndarray:
     '''simple docstring'''





     return np.array(
         [
             [gamma(_a		), -gamma(_a		) * beta(_a		), 0, 0],
             [-gamma(_a		) * beta(_a		), gamma(_a		), 0, 0],
             [0, 0, 1, 0],
             [0, 0, 0, 1],
         ]		)




def     UpperCamelCase    (      _a       ,		_a = None		)					->							np.ndarray:
     '''simple docstring'''





     if event is None:
          lowercase_       :Optional[int]     =       np.array([ct, x, y, z]		)  # Symbolic four vector
     else:
          event[0] *= c  # x0 is ct (speed of light * time)

     return transformation_matrix(_a		) @ event


if __name__ == "__main__":
   import doctest

   doctest.testmod()

   # Example of symbolic vector:
   SCREAMING_SNAKE_CASE       :		List[str]         =				transform(29_979_245)
   print("Example of four vector: ")
   print(f"ct' = {four_vector[0]}")
   print(f"x' = {four_vector[1]}")
   print(f"y' = {four_vector[2]}")
   print(f"z' = {four_vector[3]}")

   # Substitute symbols with numerical values
   SCREAMING_SNAKE_CASE       :		Optional[int]         =				{ct: c, x: 1, y: 1, z: 1}
   SCREAMING_SNAKE_CASE       :		str         =				[four_vector[i].subs(sub_dict) for i in range(4)]

   print(f"\n{numerical_vector}")

