




def lowerCamelCase__  (     _lowercase					,		_lowercase = 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
     UpperCAmelCase_     :				Tuple  =     [
         2047,
         1373653,
         25326001,
         3215031751,
         2152302898747,
         3474749660383,
         341550071728321,
         1,
         3825123056546413051,
         1,
         1,
         318665857834031151167461,
         3317044064679887385961981,
     ]

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





from functools import reduce

__a							     =					(
    '73167176531330624919225119674426574742355349194934'
    '96983520312774506326239578318016984801869478851843'
    '85861560789112949495459501737958331952853208805511'
    '12540698747158523863050715693290963295227443043557'
    '66896648950445244523161731856403098711121722383113'
    '62229893423380308135336276614282806444486645238749'
    '30358907296290491560440772390713810515859307960866'
    '70172427121883998797908792274921901699720888093776'
    '65727333001053367881220235421809751254540594752243'
    '52584907711670556013604839586446706324415722155397'
    '53697817977846174064955149290862569321978468622482'
    '83972241375657056057490261407972968652414535100474'
    '82166370484403199890008895243450658541227588666881'
    '16427171479924442928230863465674813919123162824586'
    '17866458359124566529476545682848912883142607690042'
    '24219022671055626321111109370544217506941658960408'
    '07198403850962455444362981230987879927244284909188'
    '84580156166097919133875499200524063689912560717606'
    '05886116467109405077541002256983155200055935729725'
    '71636269561882670428252483600823257530420752963450'
)







def lowerCamelCase__  (     _lowercase = N ):

     '''simple docstring'''


     return max(
         # mypy cannot properly interpret reduce
         int(reduce(lambda _lowercase					,		_lowercase : str(int(_lowercase ) * int(_lowercase ) )					,		n[i : i + 13] ) )
         for i in range(len(_lowercase ) - 12 ) )


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