**2013 APMO 2.** Determine all positive integers for which is an integer. Here denotes the greatest integer less than or equal to .

I calculated the fraction for and found that none of them satisfied the given condition:

With none of the first 16 values of satisfying the given condition, * I began to suspect that the fraction can never be an integer*.

With that in mind, I began to manipulate the given fraction into something that is easier to analyse. * When dealing with floor functions, it is often useful to introduce variables for the integer and fractional parts.* In this problem, it means writing as another variable , then expressing in terms of .

Let . Then we can write , where . Now, I can think of the problem in a slightly different manner:

*I want to find all positive integers such that the fraction is an integer for some .*

Notice that in the fraction above, the numerator has to the 4th power while the denominator only has to the 2nd power. * A common trick is to perform long division so that the numerator has lower degree than the denominator.* (When the numerator has lower degree, for large enough values of the numerator will be smaller than the denominator, and hence cannot be an integer. This helps to restrict the search space.)

Performing long division:

Hence, the original fraction is an integer if and only if is an integer.

* Remember that can only take on specific values! *In fact, when takes on its maximum value of , the fraction above becomes

hence the fraction can only take on the values of 1, 2 and 3. We can evaluate each of these cases separately.

Case 1:

Simplifying, we get

which does not have any solution in positive since no 2 consecutive squares are so close together.

Case 2:

Simplifying, we get

However, considering this equation in mod 8, we can see that there is no solution. In mod 8, all squares must be congruent to 0, 1, or 4, i.e. . However, by the same token this means that .

Case 3:

Simplifying, we get

However, considering this equation in mod 3, we can see that there is no solution.

In conclusion, there is no positive integer such that the fraction is an integer for some . Hence, there is no positive integer such that is an integer. **QED.**

(**Note:** For problems where you are supposed to find solutions to equations, it is often a good idea to consider the equation modulo for various to constrain the values which the variables can take. For example,

It’s good to remember which modulos are good for constraining the values of which powers.)