I've been looking at the following problem, involving Diophantine equations:
Let such that and . Show that there exist nonnegative integers so that .
I know that the solutions to a linear Diophantine equation are given by , where is a particular solution and is an integer parameter. If I want these to be positive simultaneously, I need to have . In other words, there needs to be an integer in the interval .
From here, I cannot seem to put bounds on the endpoints of the interval. I would appreciate any insight you may have on the issue.
Thanks. I looked into the link provided in the Yahoo answer and was pleasantly surprised to see that the proof given was geometric. I myself spent some time trying to handle the problem geometrically (with a similar approach), but I couldn't get it to work. I should be more careful next time!