How to implement an efficient infinite generator of prime numbers in Python?

Question

This is not a homework, I am just curious.

INFINITE is the key word here.

I wish to use it as for p in primes(). I believe that this is a built-

Answer 1

For posterity, here's a rewrite of Will Ness's beautiful algorithm for Python 3. Some changes are needed (iterators no longer have .next() methods, but there's a new next() builtin function). Other changes are for fun (using the new yield from replaces four yield statements in the original. More are for readability (I'm not a fan of overusing ;-) 1-letter variable names).

It's significantly faster than the original, but not for algorithmic reasons. The speedup is mostly due to removing the original's add() function, doing that inline instead.

def psieve():
    import itertools
    yield from (2, 3, 5, 7)
    D = {}
    ps = psieve()
    next(ps)
    p = next(ps)
    assert p == 3
    psq = p*p
    for i in itertools.count(9, 2):
        if i in D:      # composite
            step = D.pop(i)
        elif i < psq:   # prime
            yield i
            continue
        else:           # composite, = p*p
            assert i == psq
            step = 2*p
            p = next(ps)
            psq = p*p
        i += step
        while i in D:
            i += step
        D[i] = step