please bear with me as my mathematical abilities do not exceed good highschool level. Here is my problem:
Given is a function f(x,y). x and y are natural numbers. It is known that exactly one root exists for the function, and boundaries within which this root is located are known, too. Is there any way to find out where f(x,y)=0?
Well, if you are solving your solutions will be points on the curve (in ) . However if you are only looking for natural numbers, there usually isn't much aside from guess and check that you can do. In this case, you can factor 109. Since 109 is prime, the only pairs that will work are and .