
"""simple docstring"""


import sys



def 	lowerCamelCase__			(							UpperCAmelCase_				)->							Any:







       """simple docstring"""
       UpperCamelCase					 =		len(UpperCAmelCase_				)
       UpperCamelCase					 =		[[0 for x in range(UpperCAmelCase_				)] for x in range(UpperCAmelCase_				)]
       UpperCamelCase					 =		[[0 for x in range(UpperCAmelCase_				)] for x in range(UpperCAmelCase_				)]

       for chain_length in range(2		,      UpperCAmelCase_				):
              for a in range(1		,      n - chain_length + 1				):
                     UpperCamelCase					 =		a + chain_length - 1

                     UpperCamelCase					 =		sys.maxsize
                     for c in range(UpperCAmelCase_		,      UpperCAmelCase_				):
                            UpperCamelCase					 =		(
                                matrix[a][c] + matrix[c + 1][b] + array[a - 1] * array[c] * array[b]
                            )
                            if cost < matrix[a][b]:
                                   UpperCamelCase					 =		cost
                                   UpperCamelCase					 =		c
       return matrix, sol



def 	lowerCamelCase__			(							UpperCAmelCase_		,      UpperCAmelCase_		,      UpperCAmelCase_				)->							Optional[Any]:







       """simple docstring"""
       if i == j:
              print("A" + str(UpperCAmelCase_				)		,      end=" "				)
       else:
              print("("		,      end=" "				)
              print_optiomal_solution(UpperCAmelCase_		,      UpperCAmelCase_		,      optimal_solution[i][j]				)
              print_optiomal_solution(UpperCAmelCase_		,      optimal_solution[i][j] + 1		,      UpperCAmelCase_				)
              print(")"		,      end=" "				)



def 	lowerCamelCase__			(							)->							List[Any]:







       """simple docstring"""
       UpperCamelCase					 =		[30, 35, 15, 5, 10, 20, 25]
       UpperCamelCase					 =		len(UpperCAmelCase_				)
       # Size of matrix created from above array will be
       # 30*35 35*15 15*5 5*10 10*20 20*25
       UpperCamelCase ,       UpperCamelCase					 =		matrix_chain_order(UpperCAmelCase_				)

       print("No. of Operation required: " + str(matrix[1][n - 1]				)				)
       print_optiomal_solution(UpperCAmelCase_		,      1		,      n - 1				)


if __name__ == "__main__":
      main()







"""simple docstring"""


def 	lowerCamelCase__			(							UpperCAmelCase_		,      UpperCAmelCase_ = False				)->							bool:







       """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
       UpperCamelCase					 =		[
           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,
       ]

       UpperCamelCase					 =		[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
       for idx, _p in enumerate(UpperCAmelCase_		,      1				):
              if n < _p:
                     # then we have our last prime to check
                     UpperCamelCase					 =		primes[:idx]
                     break
       UpperCamelCase ,       UpperCamelCase					 =		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					 =		False
              for r in range(UpperCAmelCase_				):
                     UpperCamelCase					 =		pow(UpperCAmelCase_		,      d * 2**r		,      UpperCAmelCase_				)
                     # see article for analysis explanation for m
                     if (r == 0 and m == 1) or ((m + 1) % n == 0):
                            UpperCamelCase					 =		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__			(							)->							None:







       """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()





