2000 JBMO Problems/Problem 2
Problem 2
Find all positive integers such that is the square of an integer.
Solution
After rearranging we get:
Let
we get: or,
Now, it is clear from above that divides . so,
If
so
But
If then increases exponentially compared to so cannot be .
Thus .
Substituting value of above we get:
or this results in only or
Thus or .
~Kris17
Solution 2 (credit to dskull16)
n = 1 is an obvious solution but are there any more? We require that for some k in the naturals. Using difference of two squares and realising that the factor pairs can only be a power of 3, we get that which gives us . While we could consider induction on j to prove that , we could instead consider the difference between and all the powers of 3 preceding it. The smallest difference between the nth power of 3 and any other power of 3 before it is trivially the n-1th power of 3 so it suffices to show that: for , which simplifies to and hence which is trivially true . Hence there are no further solutions.