  A theorem of Gekeler compares the number of non-isomorphic automorphic
representations associated with the space of cusp forms of weight $k$ on
$\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing that the two are
equal whenever $N$ is squarefree. We prove the converse of this theorem (with
one small exception), thus providing a characterization of squarefree integers.
We also establish a similar characterization of prime numbers in terms of the
number of Hecke newforms of weight $k$ on $\Gamma_0(N)$.
  It follows that a hypothetical fast algorithm for computing the number of
such automorphic representations for even a single weight $k$ would yield a
fast test for whether $N$ is squarefree. We also show how to obtain bounds on
the possible square divisors of a number $N$ that has been found to not be
squarefree via this test, and we show how to probabilistically obtain the
complete factorization of the squarefull part of $N$ from the number of such
automorphic representations for two different weights. If in addition we have
the number of such Hecke newforms for even a single weight $k$, then we show
how to probabilistically factor $N$ entirely. All of these computations could
be performed quickly in practice, given the number(s) of automorphic
representations and modular forms as input.
