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Florian Luca

Publications and source records attributed to Florian Luca.

At least 37 records · Page 2Linked to original sources
arXiv2025

Transcendence for Pisot Morphic Words over an Algebraic Base

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $β$ such that $|β|>1$, the number $[\![\boldsymbol{u} ]\!]_β:=\sum_{n=0}^\infty \frac{u_n}{β^n}$ either lies in $\mathbb Q(β)$ or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of $[\![\boldsymbol{u}]\!]_β$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_β$ is transcendental.

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arXiv2025

Sum of Consecutive Terms of Pell and Related Sequences

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for $n\geq k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences.

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arXiv2024

Transcendence of Hecke-Mahler Series

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor nθ+α\rfloor) β^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $α$ is a real number, $θ$ is an irrational real number, and $β$ is an algebraic number such that $|β|>1$.

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arXiv2024

On the average value of the minimal Hamming multiple

We find a nontrivial upper bound on the average value of the function M(n) which associates to every positive integer n the minimal Hamming weight of a multiple of n. Some new results about the equation M(n)=M(n') are given.

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arXiv2024

On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x=L_m^{(k)}$, in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$.

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arXiv2024

On the distance between factorials and repunits

We show that if $n\ge n_0$, $b\ge 2$ are integers, $p\ge 7$ is prime and $n!-(b^p-1)/(b-1)\ge 0$, then $n!-(b^p-1)/(b-1) \ge 0.5\log\log n/\log\log\log n$. Further results are obtained, in particular for the case $n!-(b^p-1)/(b-1) < 0$.

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arXiv2024

Fibonacci--Theodorus Spiral and its properties

Inspired by the ancient spiral constructed by the greek philosopher Theodorus which is based on concatenated right triangles, we have created a spiral. In this spiral, called \emph{Fibonacci--Theodorus}, the sides of the triangles have lengths corresponding to Fibonacci numbers. Towards the end of the paper, we present a generalized method applicable to second-order recurrence relations. Our exploration of the Fibonacci--Theodorus spiral aims to address a variety of questions, showcasing its unique properties and behaviors. For example, we study topics such as area, perimeter, and angles. Notably, we establish a relationship between the ratio of two consecutive areas and the golden ratio, a pattern that extends to angles sharing a common vertex. Furthermore, we present some asymptotic results. For instance, we demonstrate that the sum of the first $n$ areas comprising the spiral approaches a multiple of the sum of the initial $n$ Fibonacci numbers. Moreover, we provide a sequence of open problems related to all spiral worked in this paper. Finally, in his work Hahn, Hahn observed a potential connection between the golden ratio and the ratio of areas between spines of lengths $\sqrt{F_{n+1}}$ and $\sqrt{F_{n+2}-1}$ and the areas between spines of lengths $\sqrt{F_{n}}$ and $\sqrt{F_{n+1}-1}$ in the Theodorus spiral. However, no formal proof has been provided in his work. In this paper, we provide a proof for Hahn's conjecture.

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arXiv2024

On the Euler function of linearly recurrence sequences

In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $ϕ(|U_n|)\ge |U_{ϕ(n)}|$ holds on a set of positive integers $n$ of density $1$, where $ϕ$ is the Euler function. In fact, we show that the set of $n\le x$ for which the above inequality fails has counting function $O_U(x/\log x)$.

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arXiv2024

Twisted rational zeros of linear recurrence sequences

We introduce the notion of a twisted rational zero of a non-degenerate linear recurrence sequence (LRS). We show that any non-degenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that $1/3$ and $-5/3$ are the only twisted rational zeros which are not integral zeros.

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arXiv2024

On a problem of Pillai involving S-units and Lucas numbers

Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases.

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arXiv2023

Multiplicative independence in the sequence of $k$-generalized Lucas numbers

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers for some fixed integer $k\ge 2$, whose first $k$ terms are $0,\;\ldots\;,\;0,\;2,\;1$ and each term afterward is the sum of the preceding $k$ terms. In this paper, we find all pairs of the $k$-generalized Lucas numbers that are multiplicatively dependent.

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arXiv2023

On the Largest Prime factor of the k-generalized Lucas numbers

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(\pm 1)=1$. We show that if $n \ge k + 1$, then $P (L_n^{(k)} ) > (1/86) \log \log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.

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arXiv2023

Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least.

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arXiv2023

On the discriminator of Lucas sequences. II

The family of Shallit sequences consists of the Lucas sequences satisfying the recurrence $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and with $k\ge 1$ arbitrary. For every fixed $k$ the integers $\{U_n(k)\}_{n\ge 0}$ are distinct, and hence for every $n\ge 1$ there exists a smallest integer $D_k(n)$, called discriminator, such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $D_k(n).$ In part I it was proved that there exists a constant $n_k$ such that $D_{k}(n)$ has a simple characterization for every $n\ge n_k$. Here, we study the values not following this characterization and provide an upper bound for $n_k$ using Matveev's theorem and the Koksma-Erdos-Turán inequality. We completely determine the discriminator $D_{k}(n)$ for every $n\ge 1$ and a set of integers $k$ of natural density $68/75$. We also correct an omission in the statement of Theorem 3 in part I.

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