On Friday, October 31, 2003, at 11:02 AM, Jon Perry wrote:
Does there always exist, for any x and y, a partition bd of ky for some k such that kx = ad+bc for some positive integers a and c?
Sylvester's coin problem is relevant here: given coins of values b and d which are relatively prime, you can make any amount of money greater than bd-b-d. (I call this the Houston Eulers problem: a football team with a perfect kicker always gets 3 or 7 points at a time, so can end the game with any score larger than 11.) You'll have to make kx larger, since you want a and c to both be positive, not just, er, zop. But if you make kx large enough, some a,c always exist, and then you can ensure positive ones by trading b d's for d b's.
Jon Perry
Jon: in 24 hours, you've just sent ten messages to this list each with at most one sentence of content. First, this borders on list abuse -- math-fun aims for high content-per-message. Second, even aside from this, your current practice will probably lead to no one reading your messages. To join the community, please pay more attention to the culture of this mailing list: think first, then post when you have something to say. --Michael Kleber kleber@brandeis.edu