I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).
We have Bezout's identity with a sign change. There are infinite solutions, and a solution exists iff x > 4 and .
For we solve for y
7*(17k + 5) - 17y = 1
7*17k + 35 - 17y = 1
17y = 7*17k + 34
y = 7k + 2
So the solution set is given by (x,y) = (5 + 17k, 2 + 7k) where k ranges over the non-negative integers.
Nothing in particular UnD; I had just skimmed over:
Diophantine Equations
and saw WB's equation...