


"""simple docstring"""

def lowerCAmelCase_      (				UpperCamelCase__				,   UpperCamelCase__ = False ):






 """simple docstring"""







 if n == 2:
  return True
 if not n % 2 or n < 2:
  return False
 if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
  return False
 if n > 3317044064679887385961981 and not allow_probable:
  raise ValueError(
      """Warning: upper bound of deterministic test is exceeded. """
      """Pass allow_probable=True to allow probabilistic test. """
      """A return value of True indicates a probable prime.""" )
 # array bounds provided by analysis
 __lowercase									=	[
     2047,
     1373653,
     25326001,
     3215031751,
     2152302898747,
     3474749660383,
     341550071728321,
     1,
     3825123056546413051,
     1,
     1,
     318665857834031151167461,
     3317044064679887385961981,
 ]

 __lowercase									=	[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
 for idx, _p in enumerate(SCREAMING_SNAKE_CASE_				,   1 ):
  if n < _p:
   # then we have our last prime to check
   __lowercase									=	primes[:idx]
   break
 __lowercase							,		__lowercase									=	n - 1, 0
 # break up n -1 into a power of 2 (s) and
 # remaining odd component
 # essentially, solve for d * 2 ** s == n - 1
 while d % 2 == 0:
  d //= 2
  s += 1
 for prime in plist:
  __lowercase									=	False
  for r in range(SCREAMING_SNAKE_CASE_ ):
   __lowercase									=	pow(SCREAMING_SNAKE_CASE_				,   d * 2**r				,   SCREAMING_SNAKE_CASE_ )
   # see article for analysis explanation for m
   if (r == 0 and m == 1) or ((m + 1) % n == 0):
    __lowercase									=	True
    # this loop will not determine compositeness
    break
  if pr:
   continue
  # if pr is False, then the above loop never evaluated to true,
  # and the n MUST be composite
  return False
 return True


def lowerCAmelCase_      (				):






 """simple docstring"""







 assert not miller_rabin(561 )
 assert miller_rabin(563 )
 # 2047

 assert not miller_rabin(838201 )
 assert miller_rabin(838207 )
 # 1_373_653

 assert not miller_rabin(17316001 )
 assert miller_rabin(17316017 )
 # 25_326_001

 assert not miller_rabin(3078386641 )
 assert miller_rabin(3078386653 )
 # 3_215_031_751

 assert not miller_rabin(1713045574801 )
 assert miller_rabin(1713045574819 )
 # 2_152_302_898_747

 assert not miller_rabin(2779799728307 )
 assert miller_rabin(2779799728327 )
 # 3_474_749_660_383

 assert not miller_rabin(113850023909441 )
 assert miller_rabin(113850023909527 )
 # 341_550_071_728_321

 assert not miller_rabin(1275041018848804351 )
 assert miller_rabin(1275041018848804391 )
 # 3_825_123_056_546_413_051

 assert not miller_rabin(79666464458507787791867 )
 assert miller_rabin(79666464458507787791951 )
 # 318_665_857_834_031_151_167_461

 assert not miller_rabin(552840677446647897660333 )
 assert miller_rabin(552840677446647897660359 )
 # 3_317_044_064_679_887_385_961_981
 # upper limit for probabilistic test


if __name__ == "__main__":
       test_miller_rabin()




"""simple docstring"""

from __future__ import annotations


def lowerCAmelCase_      (				UpperCamelCase__	: list[float] ):






 """simple docstring"""







 if len(UpperCamelCase__ ) < 2:
  raise ValueError("""Monogons and Digons are not polygons in the Euclidean space""" )
 if any(i <= 0 for i in nums ):
  raise ValueError("""All values must be greater than 0""" )
 __lowercase									=	nums.copy()
 copy_nums.sort()
 return copy_nums[-1] < sum(copy_nums[:-1] )


if __name__ == "__main__":
       import doctest

       doctest.testmod()
