## n-th number which is not a square

Find a closed form (i.e. an explicit formula in terms of ) for the -th number which is not a perfect square.

**Solution**

If we denote as being the -th number which is not a square, we can see that the following recurence holds:

Further more, we conjecture that and . We see that for we have and . We begin now to prove our conjecture by induction. Assume that . Then, using our recurence discovered earlier, we can see that , and hence . By mathematical induction we can say now that the conjecture is true always. Now, if given we can find in terms of such that , then we are almost done.

We have the following equivalences:

Therefore , where is the greatest integer not exceeding .

To find we proceed as follows:

. Replacing we get a very crowded formula which is correct.

. Simplyfying, we get

**Another closed form**

.