# 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.