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 isprime(n):
"""Naive primality test using naive and unoptimized trial division.
>>> isprime(17)
True
>>> isprime(18)
False
Naive, slow but thorough test for primality using unoptimized trial
division. This function does far too much work, and consequently is
very slow. Nevertheless, it is guaranteed to give the right answer.
Eventually.
"""
if n == 2:
return True
if n < 2 or n % 2 == 0:
return False
for i in range(3, isqrt(n)+1, 2):
if n % i == 0:
return False
return True