Search arXiv⌕ Search

arXiv subjects

Florian Luca

Publications and source records attributed to Florian Luca.

At least 19 recordsLinked to original sources
arXiv2026

On periods of the Tribonacci sequence modulo primes

In this paper, we study the $p$-adic zeros of the Tribonacci sequence $T_n$, for a prime $p$. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of $p=11$. In addition, we investigate the primes $p$ for which at the first two values $T_n$ and $T_{n-1}$ which are both divisible by $p$, both of these are actually divisible by $p^2$, proving under the assumption of a generalisation of the $abc$ conjecture that there are infinitely many $p$ which are not such.

math.NT↗
arXiv2026

On Periodic and Aperiodic Optimal Strategies in Solvency Games

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

cs.GT↗
arXiv2026

On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture

For each fixed positive integer $h$, we study the divisibility relation $σ(n)\midσ(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\log x)^2)$. We also study the proportionality equation $σ(n+h)=λσ(n)$. For every fixed nonzero integer $h$, uniformly for all real $λ>0$, the number of solutions up to $x$ is $O(x/\sqrt{\log\log\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $σ(n+1)=2σ(n)$; the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $σ(n+h)=kσ(n)$ has infinitely many positive integer solutions for every fixed $h,k\ge1$.

math.NT↗
arXiv2026

Skolem Meets Bateman-Horn

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and control theory. Decidability is notoriously open; the state of the art is a decision procedure for recurrences of order at most 4: an advance achieved some 40 years ago, based on Baker's theorem on linear forms in logarithms of algebraic numbers. A new approach to the Skolem Problem was recently initiated in [LOW21, LOW22] via the notion of a Universal Skolem Set -- a set $S$ of positive integers such that it is decidable whether a given non-degenerate linear recurrence sequence has a zero in $S$. Clearly, proving decidability of the Skolem Problem is equivalent to showing that $\mathbb{N}$ itself is a Universal Skolem Set. The main contribution of the present paper is to construct a Universal Skolem Set that has lower density at least $1/8$. We show moreover that this set has density $1$ subject to Martin's uniform formulation of the Bateman--Horn conjecture. The latter is a far-reaching quantitative hypothesis concerning the frequency of primes among the values of systems of polynomials.

cs.DM↗
arXiv2026

Multiplicative dependence in the sumset of multiplicative groups

Let $Γ$ and $Δ$ be finitely generated multiplicative groups of algebraic numbers such that $Γ\capΔ$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\in Γ$ and $y_1,y_2\in Δ$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \ \ldots, \ x_m+y_m$ with $x_1, \ldots, x_m \in Γ$ and $y_1, \ldots, y_m \in Δ$. For $m=3$ we classify such sums, up to finitely many exceptions, assuming the $abc$-conjecture.

math.NT↗
arXiv2026

Conjectural Decidability of the Skolem Problem

The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a double exponential of the magnitude of the data defining the given LRS. We establish two main results. First, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conclude, conditionally on a strengthening of the classical Cramér conjecture, that large zeros do not exist, which would entail decidability of the Skolem Problem. Second, we show unconditionally that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature.

cs.DM↗
arXiv2026

On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le \lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a, b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in\{2,3,6\}$, where $q$ is an integer which is $3$-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max\{a,b\}$. We finish with an application of the LLL algorithm to reduce this bound.

math.NT↗
arXiv2026

Asymptotic formulas for sums of elements from a multiplicative group

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $Γ$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(α)$ denote the absolute exponential height of an algebraic number $α$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\inΓ$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence.

math.NT↗
arXiv2026

Cullen and Woodall numbers in Padovan and Perrin sequences

Let $\{P_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3} = U_{n+1} + U_n$, but with initial values $P_0 = P_1 = P_2 = 1$ and $R_0 = 3$, $R_1 = 0$, $R_2 = 2$, respectively. A \textit{Cullen number} is a positive integer of the form $m\cdot 2^m + 1$ for some integer $m \ge 1$, while a \textit{Woodall number} is a positive integer of the form $m\cdot 2^m - 1$ for some integer $m \ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence.

math.NT↗
arXiv2026

On generalized Thabit numbers $(p+1)p^\mathfrak{a}-1$ in the $k$-Lucas sequence

Let $k\ge 2$ and $\{L_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$-Lucas numbers 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 solve the Diophantine equation $L_n^{(k)}=(p+1)p^\mathfrak{a}-1$, for a Mersenne or Fermat prime $p=2^{\ell}\pm 1$, and positive integers $n\ge 2$, $k\ge 2$, $\mathfrak{a}\ge 1$ and $\ell \ge 1$.

math.NT↗
arXiv2026

Product of powers of distinct primes as sums of Fibonacci numbers

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes $(q,p)$ with $q\le \min\{1000,p\}$ such that the above equation has at least two solutions $(x,y)$ (and corresponding $m,n$) in positive integers.

math.NT↗
arXiv2026

On the number of divisors of Mersenne numbers

Denote $f(n):=\sum_{1\le k\le n} τ(2^k-1)$, where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold.

math.NT↗
arXiv2026

Poor man's transcendence for Frobenius traces of elliptic curves

Let $E$ be an elliptic curve without complex multiplication defined over $\mathbb Q$. Viewing the sequence of its Frobenius traces $(a_p(E))_p$ indexed by primes $p$ as an element in the "poor man's adèle ring", we prove its transcendence over $\mathbb Q$.

math.NT↗
arXiv2025

Concatenations of Terms of an Arithmetic Progression

Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$ terms of $(u(n))_{n\in\mathbb{N}}$ in base $b$ from the right; $s_g(n) = \overline{u(n)u(n-1)\cdots u(0)}^b$; and $(s_*(n))_{n\in\mathbb{N}}$, given by $s_*(0)=u(0)$, $s_*(n)=\overline{s(n)s_g(n-1)}^b, n\geq 1$. We construct explicit formulae for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented $(s(n))_{n\in\mathbb{N}}$ and $(s_g(n))_{n\in\mathbb{N}}$ in the decimal base when $(u(n))_{n\in\mathbb{N}}=\mathbb{N}\setminus \{0\}$.

math.CO↗
arXiv2025

On the growth of hypergeometric sequences

Hypergeometric sequences obey first-order linear recurrence relations with polynomial coefficients and are commonplace throughout the mathematical and computational sciences. For certain classes of hypergeometric sequences, we prove linear growth estimates on their Weil heights. We give an application of our effective results towards the Membership Problem from Computer Science. Recall that Membership asks to procedurally determine whether a specified target is an element of a given recurrence sequence.

math.NT↗
arXiv2025

On the Decidability of Presburger Arithmetic Expanded with Powers

We prove that for any integers $α, β> 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, α^{\mathbb{N}}, β^{\mathbb{N}}\rangle$ is decidable (where $α^{\mathbb{N}}$ is the set of positive integer powers of $α$, and likewise for $β^{\mathbb{N}}$). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of $\langle \mathbb{N}; 0,1, <, +, x\mapsto α^x, x \mapsto β^x\rangle$ for any multiplicatively independent $α,β> 1$ would lead to mathematical breakthroughs regarding base-$α$ and base-$β$ expansions of certain transcendental numbers.

cs.LO↗
arXiv2025

Irrationality and transcendence questions in the "poor man's adèle ring"

We discuss arithmetic questions related to the "poor man's adèle ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main theorem is about the $\mathcal A$-transcendence of the element $(F_p(q))_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.

math.NT↗