  This work brings techniques from the theory of recurrent integer sequences to
the problem of balancedness of symmetric Boolean functions. In particular, the
periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean
functions is considered. Periods modulo $p$, bounds for periods and relations
between them are obtained for these exponential sums. The concept of avoiding
primes is also introduced. This concept and the bounds presented in this work
are used to show that some classes of symmetric Boolean functions are not
balanced. In particular, every elementary symmetric Boolean function of degree
not a power of 2 and less than 2048 is not balanced. For instance, the
elementary symmetric Boolean function in $n$ variables of degree $1292$ is not
balanced because the prime $p = 176129$ does not divide its exponential sum for
any positive integer $n$. It is showed that for some symmetric Boolean
functions, the set of primes avoided by the sequence of exponential sums
contains a subset that has positive density within the set of primes. Finally,
in the last section, a brief study for the set of primes that divide some term
of the sequence of exponential sums is presented.
