2005 JBMO Problems/Problem 1
Find all positive integers satisfying the equation
We can re-write the equation as:
The above equation tells us that is a perfect square. Since . this implies that
Also, taking on both sides we see that cannot be a multiple of . Also, note that has to be even since is a perfect square. So, cannot be even, implying that is odd.
So we have only to consider for .
Trying above 5 values for we find that result in perfect squares.
Thus, we have cases to check:
Thus all solutions are and .