## Squarefree Hilbert numbers

### Problem 580

A **Hilbert number** is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a *squarefree Hilbert number* as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times13$. However $6237$ is a Hilbert number that is not squarefree in this sense, as it is divisible by $9^2$. The number $3969$ is also not squarefree, as it is divisible by both $9^2$ and $21^2$.

There are $2327192$ squarefree Hilbert numbers below $10^7$.

How many squarefree Hilbert numbers are there below $10^{16}$?