Problem:
Find the number of ordered pairs of positive integer solutions to the equation
Solution:
Note that if and only if , so it is sufficient to find the number of positive integer solutions to the second equation. One positive integer solution to this equation is . If is another such solution, then , so for some integer and . Thus the solutions to have the form , and substituting these pairs into the original equation shows that they are all solutions. Both of these values are positive if and only if . Thus there are solutions with positive integer values of the original equation.
The problems on this page are the property of the MAA's American Mathematics Competitions