2005 JBMO Problems/Problem 1
Problem 1
Find all positive integers satisfying the equation
Solution
We can re-write the equation as:
or
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
.