arXiv subjects
Publications and source records attributed to Ingrid Vukusic.
Let $a\in \mathbb{Z}$ and $ρ$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=\mathbb{Q}(ρ)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2\in \mathbb{Z}[ρ]^*$ and $n\in \mathbb{Z}$. We completely solve the unit equations under the restriction $|n|\leq \max\{1,|a|^{1/3}\}$.
In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.
This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are $(2,8)$ and $(3,9)$. For triples, we prove the following theorem: If $a\notin \{2,8\}$ is a fixed integer larger than 1, then there are only finitely many triples $(a,b,c)$ of pairwise distinct integers larger than 1 such that $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ are each multiplicatively dependent. Moreover, these triples can be determined effectively.
Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In particular, the density of such integers is $0$.