arXiv subjects
Publications and source records attributed to Hemar Godinho.
We establish two new Waring--Goldbach type representations: every sufficiently large odd integer $n$ can be expressed as \[ n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c, \] where each $p_i$ is prime and $c \in \{6,7\}$.
In this paper, we define the constant $D(φ, p)$, an analogue for the Davenport constant, for sequences on the finite field $\mathbb{F}_p$, defined via quadratic symmetric polynomials. Next, we state a series of results presenting either the exact value of $D(φ, p)$, or lower and upper bounds for this constant.
In this paper we present a new formula for the number of unrestricted partitions of $n$. We do this by introducing a correspondence between the number of unrestrited partitions of $n$ and the number of non-negative solutions of systems of two equations, involving natural numbers in the interval (1 $,n^{2}$).
Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length $n$ and $η_A(G)$ defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $α\geq β$, and $A=\left\{x\in\mathbb{N}\; : \; 1 \le a \le p^α \; \mbox{ and }\; \gcd(a, p) = 1\right \}$, we have $s_{A}(\mathbb{Z}_{p^α}\oplus \mathbb{Z}_{p^β}) = η_A(\mathbb{Z}_{p^α}\oplus \mathbb{Z}_{p^β}) + p^α-1 = p^α + α+β$ and classify all the extremal $A$-weighted zero-sum free sequences.
Let $C_n$ be the cyclic group of order $n$ and set $s_{A}(C_n^r)$ as the smallest integer $\ell$ such that every sequence $\mathcal{S}$ in $C_n^r$ of length at least $\ell$ has an $A$-zero-sum subsequence of length equal to $\exp(C_n^r)$, for $A=\{-1,1\}$. In this paper, among other things, we give estimates for $s_A(C_3^r)$, and prove that $s_A(C_{3}^{3})=9$, $s_A(C_{3}^{4})=21$ and $41\leq s_A(C_{3}^{5})\leq45$.