[The combination of the quoting convention I've been using and the quoting convention Bill's been using is incredibly confusing...] Bill Gosper wrote:
Well, we *are* looking for the "greatest".-) Do you propose a different answer, or deny there is one?
I (Gareth) replied:
I deny there is one, or at least that there is any number that has a good claim.
And Bill replied:
Please name one.
No, I *deny* that there is any number that has a good claim, so of course I can't name one. I'm sorry if I wasn't clear enough. To me, the gcd is defined by properties like these: "gcd(a,b) divides both a and b and is a multiple of anything else that does." "gcd(a,b) is the single generator of the (fractional) ideal generated by a,b." Zero doesn't have any of the right properties to be the gcd of 1 and pi. Nothing does. * Bill:
Touché. New wording: GCD(a,b):= the limit of the Euclidean process of iteratively subtracting the smaller from the larger.
Gareth:
The *limit* of doing this for commensurable quantities is zero. The gcd is the last thing you get immediately before 0.
Bill:
n[755]:= NestList[ Simplify[{Max[#] - Min[#], Min[#]}] &, {GoldenRatio - 1/2, 1/√5}, 6]
Out[755]= {{-(1/2) + GoldenRatio, 1/√5}, {3/(2 √5), 1/√5}, {1/(2 √5), 1/√5}, {1/(2 √5), 1/(2 √5)}, {0, 1/(2 √5)}, {1/(2 √5), 0}, {1/(2 √5), 0}}
So GCD[GoldenRatio - 1/2, 1/√5] is 1/(2√5)
Oh, OK, so you mean the limit of the larger one. Fair enough. (I still don't see that when you have a process that in "uncontroversial" cases terminates after finitely many steps it's necessarily right to say that in cases where it approaches a limit that's the right value to use.) -- g