A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight k on Γ₀(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 Γ₀(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. This is joint work with Greg Martin.