




def       UpperCAmelCase__  (     lowerCamelCase   ):
  if divisor % 5 == 0 or divisor % 2 == 0:
    return 0
  lowercase    :str      	=						1
  lowercase    :Tuple      	=						1
  while repunit:
    lowercase    :Dict      	=						(10 * repunit + 1) % divisor
    repunit_index += 1
  return repunit_index



def       UpperCAmelCase__  (     lowerCamelCase = 1000000   ):
  lowercase    :List[Any]      	=						limit - 1
  if divisor % 2 == 0:
    divisor += 1
  while least_divisible_repunit(lowerCamelCase   ) <= limit:
    divisor += 2
  return divisor


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









def       UpperCAmelCase__  (     lowerCamelCase,			lowerCamelCase = False   ):
  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    :Optional[Any]      	=						[
      2047,
      1373653,
      25326001,
      3215031751,
      2152302898747,
      3474749660383,
      341550071728321,
      1,
      3825123056546413051,
      1,
      1,
      318665857834031151167461,
      3317044064679887385961981,
  ]

  lowercase    :List[Any]      	=						[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
  for idx, _p in enumerate(lowerCamelCase,			1   ):
    if n < _p:
      # then we have our last prime to check
      lowercase    :Tuple      	=						primes[:idx]
      break
  lowercase	,    lowercase    :str      	=						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    :Tuple      	=						False
    for r in range(lowerCamelCase   ):
      lowercase    :Any      	=						pow(lowerCamelCase,			d * 2**r,			lowerCamelCase   )
      # see article for analysis explanation for m
      if (r == 0 and m == 1) or ((m + 1) % n == 0):
        lowercase    :Any      	=						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       UpperCAmelCase__  (     ):
  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()



