




def       _SCREAMING_SNAKE_CASE      (							lowercase     :	int     ,							lowercase     :	bool = 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 > 3_31_70_44_06_46_79_88_73_85_96_19_81 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
      lowerCamelCase_							=   [
          20_47,
          1_37_36_53,
          25_32_60_01,
          32_15_03_17_51,
          2_15_23_02_89_87_47,
          3_47_47_49_66_03_83,
          3_41_55_00_71_72_83_21,
          1,
          3_82_51_23_05_65_46_41_30_51,
          1,
          1,
          31_86_65_85_78_34_03_11_51_16_74_61,
          3_31_70_44_06_46_79_88_73_85_96_19_81,
      ]

      lowerCamelCase_							=   [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
                  lowerCamelCase_							=   primes[:idx]
                  break
      lowerCamelCase_					,    lowerCamelCase_							=   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:
            lowerCamelCase_							=   False
            for r in range(lowercase					):
                  lowerCamelCase_							=   pow(lowercase     ,							d * 2**r     ,							lowercase					)
                  # see article for analysis explanation for m
                  if (r == 0 and m == 1) or ((m + 1) % n == 0):
                        lowerCamelCase_							=   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       _SCREAMING_SNAKE_CASE      (							):


      '''simple docstring'''



      assert not miller_rabin(5_61					)
      assert miller_rabin(5_63					)
      # 2047

      assert not miller_rabin(83_82_01					)
      assert miller_rabin(83_82_07					)
      # 1_373_653

      assert not miller_rabin(17_31_60_01					)
      assert miller_rabin(17_31_60_17					)
      # 25_326_001

      assert not miller_rabin(30_78_38_66_41					)
      assert miller_rabin(30_78_38_66_53					)
      # 3_215_031_751

      assert not miller_rabin(1_71_30_45_57_48_01					)
      assert miller_rabin(1_71_30_45_57_48_19					)
      # 2_152_302_898_747

      assert not miller_rabin(2_77_97_99_72_83_07					)
      assert miller_rabin(2_77_97_99_72_83_27					)
      # 3_474_749_660_383

      assert not miller_rabin(1_13_85_00_23_90_94_41					)
      assert miller_rabin(1_13_85_00_23_90_95_27					)
      # 341_550_071_728_321

      assert not miller_rabin(1_27_50_41_01_88_48_80_43_51					)
      assert miller_rabin(1_27_50_41_01_88_48_80_43_91					)
      # 3_825_123_056_546_413_051

      assert not miller_rabin(7_96_66_46_44_58_50_77_87_79_18_67					)
      assert miller_rabin(7_96_66_46_44_58_50_77_87_79_19_51					)
      # 318_665_857_834_031_151_167_461

      assert not miller_rabin(55_28_40_67_74_46_64_78_97_66_03_33					)
      assert miller_rabin(55_28_40_67_74_46_64_78_97_66_03_59					)
      # 3_317_044_064_679_887_385_961_981
      # upper limit for probabilistic test


if __name__ == "__main__":
  test_miller_rabin()









def       _SCREAMING_SNAKE_CASE      (							lowercase     :	Tuple     ,							lowercase     :	Dict     ,							lowercase     :	List[str]     ,							lowercase     :	Dict     ,							lowercase     :	Dict     ,							lowercase     :	List[str]					):


      '''simple docstring'''



      if index == r:
            for j in range(lowercase					):
                  print(data[j]     ,							end=' '					)
            print(' '					)
            return
      #  When no more elements are there to put in data[]
      if i >= n:
            return
      # current is included, put next at next location
      lowerCamelCase_							=   arr[i]
      combination_util(lowercase     ,							lowercase     ,							lowercase     ,							index + 1     ,							lowercase     ,							i + 1					)
      # current is excluded, replace it with
      # next (Note that i+1 is passed, but
      # index is not changed)
      combination_util(lowercase     ,							lowercase     ,							lowercase     ,							lowercase     ,							lowercase     ,							i + 1					)
      # The main function that prints all combinations
      # of size r in arr[] of size n. This function
      # mainly uses combinationUtil()

def       _SCREAMING_SNAKE_CASE      (							lowercase     :	int     ,							lowercase     :	Any     ,							lowercase     :	Tuple					):


      '''simple docstring'''



      lowerCamelCase_							=   [0] * r
      # Print all combination using temporary array 'data[]'
      combination_util(lowercase     ,							lowercase     ,							lowercase     ,							0     ,							lowercase     ,							0					)


if __name__ == "__main__":
 # Driver code to check the function above
 lowerCamelCase    :		int			   =       [10, 20, 30, 40, 50]
 print_combination(arr, len(arr), 3)
 # This code is contributed by Ambuj sahu



