Y. Puri, T. Ward
The only recurrence sequence satisfying the Fibonacci recurrence and realizable as the number of periodic points of a map is (a multiple of) the Lucas sequence.
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We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with $0 \leq n_{i} \leq p-1,n,n_{i}\in\mathbb{N}$. In this note, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.
Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property
本文证明了 Jacobsthal 和 Jacobsthal-Lucas 数列中不存在具有 Lehmer 性质的数。
A composite positive integer n is Lehmer if ϕ(n) divides n-1, where ϕ(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no Lehmer number in the Fibonacci sequence. In this paper, we adapt the method from [7] to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence $(L_n)_{n\geq 0}$ given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ for all $n\geq 0.$
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive te…
We study Gibonacci sequences mod $m$, giving special attention to the Lucas numbers. It is known which $m$ have the property that the Fibonacci sequence contains all residues mod $m$. When $m$ has this property, we say that the Fibonacci sequence is complete mod $m$. We extend this work to all Gibonacci sequences, concluding by determining the set of $m$ in which a generic Gibonacci sequence containing the relatively prime consecutive terms $a, b$ is complete mod $m$.
On the Lucas and Lehmer sequences in Dedekind domains
该研究将 Lucas 与 Lehmer 序列的强整除性推广到 Dedekind 整环，并在函数域中证明相应的 Zsigmondy 型定理与本原素因子结果。
The purpose of this paper is to investigate integer sequences with exponent lifting property, a property common in Fibonacci or Lucas sequences.
On a Diophantine Equation Involving Lucas Numbers
本文研究了涉及卢卡斯数的丢番图方程，证明了在特定条件下该方程无解，采用了因式分解、数论引理和计算机辅助计算等方法。
Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that U_n(P,Q) is a perfect square. The arguments are elementary. The main part of the paper is devoted to finding all Lucas sequences such that U_8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a ``Q-rationality'' condition on the X-coordi…
We consider the family of Lucas sequences uniquely determined by $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the discriminator function $\mathcal{D}_k(n)$ of $U_n(k)$ is defined as the smallest integer $m$ such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $m$. Numerical work of Shallit on $\mathcal{D}_k(n)$ suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every $k\ge 1$ there is a co…
Cancellation profiles of Fibonacci-Lucas zeta sums
研究Fibonacci-Lucas zeta和的项量级分布，证明归一化绝对项和的指数展开及收敛性，并确定最大项索引与条件数增长。
$p$-Linear schemes for sequences modulo $p^r$
本文研究模素数幂 $p^r$ 下组合序列的复杂行为，构造了描述该行为的 $p$-线性方案，并证明了对于固定 $r$，状态数上界不依赖于素数 $p$。
Given a positive integer $M$ and $q \in (1, M+1]$ we consider expansions in base $q$ for real numbers $x \in \left[0, {M}/{q-1}\right]$ over the alphabet $\{0, \ldots, M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\mathbf{V}_q, σ)$ related to unique expansions in such base $q$. We characterise the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, σ)$ has the specification property and the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, σ)$ is a synchronised subshift. Such properties are studied by analysing the combinatorial and dynamical propertie…
In 1982, Gessel showed that the Apéry numbers associated to the irrationality of $ζ(3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol--van Straten and Rowland--Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $(η)$ we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist--Zudilin numbers are periodic modulo …
Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum
本文针对一类含多项式与次数不小于2的约束齐次线性递推序列的广义Ledin-Brousseau型有限和，推导并给出了其存在性与唯一性性质。
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