In the theory of subgroup growth, properties of the number of subgroups of given index in a given (infinite) group are investigated. In this talk, we are interested in the number of free subgroups of given index in PSL_2(Z), and we investigate their behaviour modulo powers of primes. Since the generating function for these numbers satisfies
a Riccati differential equation, this will amount to the study of the power series solution of this differential
equation modulo prime powers. In the end, it will turn out that these numbers are always periodic modulo powers
$p^\alpha$, for primes $p$ at least $5$. I shall describe the (experimental) path to this result, which - as I believe - is quite interesting.