





from collections.abc import Callable

import numpy as np






def 							_lowerCAmelCase (  _a   :					Callable					,   _a   :					float					,   _a   :					float					,   _a   :					float					,   _a   :					float			)	->			np.ndarray:
     lowerCAmelCase_					:       str    =		int(np.ceil((x_end - xa) / step_size			)			)
     lowerCAmelCase_					:       List[str]    =		np.zeros((n + 1,)			)
     lowerCAmelCase_					:       Dict    =		ya
     lowerCAmelCase_					:       Tuple    =		xa

     for k in range(_a			):
          lowerCAmelCase_					:       Optional[Any]    =		y[k] + step_size * ode_func(_a					,   y[k]			)
          x += step_size

     return y


if __name__ == "__main__":
      import doctest

      doctest.testmod()







def 							_lowerCAmelCase (  _a   :					int					,   _a   :					bool = False			)	->			bool:
     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_					:       Optional[int]    =		[
         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_					:       Dict    =		[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
     for idx, _p in enumerate(_a					,   1			):
          if n < _p:
               # then we have our last prime to check
               lowerCAmelCase_					:       str    =		primes[:idx]
               break
     lowerCAmelCase_   ,  lowerCAmelCase_					:       Optional[Any]    =		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_					:       Tuple    =		False
          for r in range(_a			):
               lowerCAmelCase_					:       Tuple    =		pow(_a					,   d * 2**r					,   _a			)
               # see article for analysis explanation for m
               if (r == 0 and m == 1) or ((m + 1) % n == 0):
                    lowerCAmelCase_					:       List[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 							_lowerCAmelCase (  )	->			None:
     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()
