Dennis Sweeney <sweeney.dennis...@gmail.com> added the comment:
I think the behavior of gcd() == 0 is correct, but it should be documented,
because it isn't completely obvious.
Arguments for gcd() == 0:
- Preserves the invariant gcd(itertools.chain(iterables)) ==
gcd(itertools.starmap(gcd, iterables)) in the case that some iterables are
empty.
- Return a "neutral element" (not technically true for negative integers
because gcd(0, a) == abs(a) != a, but it does get "ignored")
Other considerations:
- The "common divisors" of the empty set are the integers d such that for all x
in the empty set, d divides x, which is to say, all of the integers. Then there
is no greatest of these common divisors. The same is true if we try gcd(0, 0,
... 0).
- The above is only fixed if we interpret "greatest" in the sense of
divisibility, where 0 > everything because everything divides 0. In particular,
we can say that g is a greatest common divisor of a_1, ..., a_n if g | a_i for
all i, and for any d with d | a_i for all i, we also have d | g. Using this
definition also requires that we specify that if there are two gcds, we return
the positive one.
I believe this is the correct interpretation, but it may not be obvious, since
it assumes that proof that such a number even exists.
- Another common definition of the greatest common divisor of {a_1, ..., a_n}
is the smallest positive integer expressible as an integer linear combination
x_1 a_1 + ... + x_1 a_2, where x_i are integers. But this definition breaks on
gcd(), gcd(0), gcd(0, 0), etc, since no positive integers are expressible.
In short, while I agree with the behavior, I think there should be a clarifying
sentence in the docs saying "returns 0 if all of the arguments are 0".
----------
nosy: +Dennis Sweeney
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Python tracker <rep...@bugs.python.org>
<https://bugs.python.org/issue39648>
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