
'''simple docstring'''





import math





def     snake_case							(						UpperCAmelCase		)->     bool:







      """simple docstring"""

      if 1 < number < 4:
            # 2 and 3 are primes
            return True
      elif number < 2 or number % 2 == 0 or number % 3 == 0:
            # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
            return False

      # All primes number are in format of 6k +/- 1
      for i in range(5 ,					int(math.sqrt(UpperCAmelCase		) + 1		) ,					6		):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True





def     snake_case							(						UpperCAmelCase = 1_0_0_0 ,					UpperCAmelCase = 1_0_0_0		)->     int:







      """simple docstring"""
      __A    		=     [0, 0, 0]  # length, a, b
      for a in range((a_limit * -1) + 1 ,					UpperCAmelCase		):
            for b in range(2 ,					UpperCAmelCase		):
                  if is_prime(UpperCAmelCase		):
                        __A    		=     0
                        __A    		=     0
                        while is_prime((n**2) + (a * n) + b		):
                              count += 1
                              n += 1
                        if count > longest[0]:
                              __A    		=     [count, a, b]
      __A    		=     longest[1] * longest[2]
      return ans


if __name__ == "__main__":
 print(solution(1_0_0_0, 1_0_0_0))








'''simple docstring'''





from __future__ import annotations





def     snake_case							(						UpperCAmelCase ,					UpperCAmelCase ,					UpperCAmelCase		)->     tuple[float, float]:







      """simple docstring"""
      assert (
          c % greatest_common_divisor(UpperCAmelCase ,					UpperCAmelCase		) == 0
      )  # greatest_common_divisor(a,b) function implemented below
      ((__A)		, (__A)		, (__A))    		=     extended_gcd(UpperCAmelCase ,					UpperCAmelCase		)  # extended_gcd(a,b) function implemented below
      __A    		=     c / d
      return (r * x, r * y)





def     snake_case							(						UpperCAmelCase ,					UpperCAmelCase ,					UpperCAmelCase ,					UpperCAmelCase = 2		)->     None:







      """simple docstring"""
      ((__A)		, (__A))    		=     diophantine(UpperCAmelCase ,					UpperCAmelCase ,					UpperCAmelCase		)  # Initial value
      __A    		=     greatest_common_divisor(UpperCAmelCase ,					UpperCAmelCase		)
      __A    		=     a // d
      __A    		=     b // d

      for i in range(UpperCAmelCase		):
            __A    		=     xa + i * q
            __A    		=     ya - i * p
            print(UpperCAmelCase ,					UpperCAmelCase		)





def     snake_case							(						UpperCAmelCase ,					UpperCAmelCase		)->     int:







      """simple docstring"""
      if a < b:
            __A		, __A    		=     b, a

      while a % b != 0:
            __A		, __A    		=     b, a % b

      return b





def     snake_case							(						UpperCAmelCase ,					UpperCAmelCase		)->     tuple[int, int, int]:







      """simple docstring"""
      assert a >= 0 and b >= 0

      if b == 0:
            __A		, __A		, __A    		=     a, 1, 0
      else:
            ((__A)		, (__A)		, (__A))    		=     extended_gcd(UpperCAmelCase ,					a % b		)
            __A    		=     q
            __A    		=     p - q * (a // b)

      assert a % d == 0 and b % d == 0
      assert d == a * x + b * y

      return (d, x, y)


if __name__ == "__main__":
 from doctest import testmod

 testmod(name="diophantine", verbose=True)
 testmod(name="diophantine_all_soln", verbose=True)
 testmod(name="extended_gcd", verbose=True)
 testmod(name="greatest_common_divisor", verbose=True)






