def primes1():
"""Generate prime numbers by trial division very slowly.
>>> p = primes1()
>>> [next(p) for _ in range(10)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
This adds a single optimization to ``primes0``, using a short-circuit
test for primality: as soon as a factor is found, the candidate is
rejected immediately.
"""
i = 2
yield i
while True:
i += 1
if all(i%p != 0 for p in range(2, i)):
yield i
def primes2():
"""Generate prime numbers by trial division very slowly.
>>> p = primes2()
>>> [next(p) for _ in range(10)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
This is an incremental improvement over ``primes1`` by only testing
odd numbers as potential primes and factors.
"""
yield 2
i = 3
yield i
while True:
i += 2
if all(i%p != 0 for p in range(3, i, 2)):
yield i
def primes3():
"""Generate prime numbers by trial division slowly.
>>> p = primes3()
>>> [next(p) for _ in range(10)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
This is an incremental improvement over ``primes2`` by only testing
potential factors up to the square root of the candidate. For small
primes below 50000 or so, this may be slightly faster than ``primes4``.
"""
yield 2
i = 3
yield i
while True:
i += 2
if all(i%p != 0 for p in range(3, isqrt(i)+1, 2)):
yield i
def test_all(self):
self.assertTrue(compat23.all([1, 2, 3, 4]))
self.assertTrue(compat23.all([]))
self.assertFalse(compat23.all([1, 2, 0, 4]))
def test_all(self):
self.assertTrue(compat23.all([1, 2, 3, 4]))
self.assertTrue(compat23.all([]))
self.assertFalse(compat23.all([1, 2, 0, 4]))
