In this paper we compute the exact 2-divisibility of exponential sums associated to elementary symmetric Boolean functions.
Our computation gives an affirmative answer to most of the open boundary cases of Cusick-Li-Stanica's conjecture. As a byproduct,
we prove that the 2-divisibility of these families satisfies a linear recurrence. In particular, we provide a new elementary
method to compute 2-divisibility of symmetric Boolean functions.
