>>> from prime import improved_iterations_is_prime

>>> improved_iterations_is_prime(1)
(False, 0)
>>> improved_iterations_is_prime(2)
(True, 0)
>>> improved_iterations_is_prime(53)
(True, 3)
>>> improved_iterations_is_prime(63)
(False, 1)
>>> improved_iterations_is_prime(71)
(True, 3)
>>> improved_iterations_is_prime(84)
(False, 1)
>>> improved_iterations_is_prime(97)
(True, 4)
>>> improved_iterations_is_prime(125)
(False, 2)