Find all n∈in such that the number of digits of n equal to n

Note that , , , and so on. The function is continuous and increasing for all , so when , we have ; when , ; and so on.

This means



which means we can capture the number of digits of with the function .

So the problem is the same as finding positive integer solutions to



We know that has one digit, so clearly this must be a solution. We need to show that this is the only solution.

Recall that , while . This means increases at a much slower rate than as . We know the two functions intersect when . Therefore it's clear that for all .

Now, it's always the case that , so we're essentially done:



which means there are no other solutions than .