




'''simple docstring'''


from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
__lowercase							:   str					=     299792458

# Symbols
__lowercase							:   int					=     symbols('''ct x y z''')




def lowercase_	(   _lowercase				)     ->							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 lowercase_	(   _lowercase				)     ->							float:
					'''simple docstring'''

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




def lowercase_	(   _lowercase				)     ->							np.ndarray:
					'''simple docstring'''

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




def lowercase_	(   _lowercase			,       _lowercase = None				)     ->							np.ndarray:
					'''simple docstring'''

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

					return transformation_matrix(_lowercase				) @ event


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					# Example of symbolic vector:
					__lowercase							:   int					=     transform(29979245)
					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
					__lowercase							:   Dict					=     {ct: c, x: 1, y: 1, z: 1}
					__lowercase							:   Optional[Any]					=     [four_vector[i].subs(sub_dict) for i in range(4)]

					print(f'\n{numerical_vector}')




'''simple docstring'''


import itertools
import math




def lowercase_	(   _lowercase				)     ->							bool:
					'''simple docstring'''

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or number % 2 == 0 or number % 3 == 0:
										# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
										return False

					# All primes number are in format of 6k +/- 1
					for i in range(5			,       int(math.sqrt(_lowercase				) + 1				)			,       6				):
										if number % i == 0 or number % (i + 2) == 0:
															return False
					return True




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

					lowerCamelCase_					:      Union[str, Any]     =		2
					while True:
										if is_prime(_lowercase				):
															yield num
										num += 1




def lowercase_	(   _lowercase = 10_001				)     ->							int:
					'''simple docstring'''

					return next(itertools.islice(prime_generator()			,       nth - 1			,       _lowercase				)				)


if __name__ == "__main__":
					print(f'{solution() = }')


