No abstract

No abstract

Let n > 1 be an integer, and let F p denote a field of p elements for a prime p ≡ 1 (mod n). By 2015, the question of existence or nonexistence of n-th power residue difference sets in F p had been settled for all n < 24. We settle the case n = 24 by proving the nonexistence of 24-th power residue difference sets in F p . We also prove the nonexistence of qualified 24-th power residue difference sets in F p . The proofs make use of a Mathematica program which computes formulas for the cyclotomic numbers of order 24 in terms of parameters occurring in quadratic partitions of p.

Let $S(n,k)$ denote the $n \times n$ symmetric Toeplitz band matrix whose first $k$ superdiagonals and first $k$ subdiagonals have all entries $1$, and whose remaining entries are all $0$. For all $n > k >0$ with $k$ even, we give formulas for the nullity of $S(n,k)$. As an application, it is shown that over half of these matrices $S(n,k)$ are nonsingular. For the purpose of rapid computation, we devise an algorithm that quickly computes the nullity of $S(n,k)$ even for extremely large values of $n$ and $k$, when $k$ is even. The algorithm is based on a connection between the nullspace vectors of $S(n,k)$ and the cycles in a certain directed graph.

For all n > k ≥ 1, we give formulas for the nullity N (n, k) of the n × n skew-symmetric Toeplitz band matrix whose first k superdiagonals have all entries 1 and whose remaining superdiagonals have all entries 0. This is accomplished by counting the number of cycles in certain directed graphs. As an application, for each fixed integer z ≥ 0 and large fixed k, we give an asymptotic formula for the percentage of n > k satisfying N (n, k) = z. For the purpose of rapid computation, an algorithm is devised that quickly computes N (n, k) even for extremely large values of n and k.