Lower bounds are obtained on the degree and the number of monomials of
Boolean functions, considered as a polynomial over $GF(2)$,
which decide if a given $r$-bit integer is square-free.
Similar lower bounds are also obtained for polynomials
over the reals which provide a threshold representation
of the above Boolean functions. These results provide first non-trivial lower bounds on
the complexity of a number theoretic
problem which is closely related to the integer factorization problem.