In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than 2 n .
The sequence {x n } defined by x n = (n + x n−1 )/(1 − nx n−1 ), with x 1 = 1, appeared in the context of some arctangent sums. We establish the fact that x n = 0 for n 4 and conjecture that x n is not an integer for n 5. This conjecture is given a combinatorial interpretation in terms of Stirling numbers via the elementary symmetric functions. The problem features linkage with a well-known conjecture on the existence of infinitely many primes of the form n 2 + 1, as well as our conjecture that (1 + 1 2 )(1 + 2 2 ) · · · (1 + n 2 ) is not a square for n > 3. We present an algorithm that verifies the latter for n 10 3200 .
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-Stǎnicǎ'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.
Abstract. Let t n be a sequence that satisfies a first order homogeneous recurrence t n = Q(n)t n−1 , where Q is a polynomial with integer coefficients. We describe the asymptotic behavior of the p-adic valuation of t n .
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