arXiv subjects

Armand Noubissie

Publications and source records attributed to Armand Noubissie.

9 recordsLinked to original sources

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

Large common values of generalized Ankeny-Brauer-Chowla recurrences

In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these sequences have at most two sufficiently large common values. Our proof combines Baker's theory of linear forms in logarithms of algebraic numbers with techniques from function field theory and from Galois theory.

math.NT↗

arXiv2025

Quantitative growth of multi-recurrence sequences

In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we prove a quantitative version of that result by giving an explicit upper bound for the maximal possible growth rate of a multi-recurrence. Moreover, we also give a function field analogue of the result, answering a question posed by Fuchs and Heintze when proving a bound on the growth of multi-recurrences in number fields.

math.NT↗

arXiv2025

Decidability of multiplicative matrix equations and related Diophantine problems

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains as presented in the recent book of Evertse and Györy. The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also works for systems of symmetric matrices which do not form a semigroup. In the final section some related counting problems are investigated.

math.NT↗

arXiv2025

On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$

Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}>1.$ First, we correct an oversight from \cite{X-Z}. Then, we show that the equation in the title with $b \equiv 3 \pmod 8$, $b$ prime and $a$ even has no solution in positive integers $n, x$. This generalizes a result of Szalay \cite{L}.

math.NT↗

arXiv2025

Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root

Here, we show that if $u_n=n2^n\pm 1$, then the largest prime factor of $u_n\pm m!$ for $n\ge 0,~m\ge 2$ tends to infinity with $\max\{m,n\}$. In particular, the largest $n$ participating in the equation $u_n\pm m!=2^a3^b5^c7^d$ with $n\ge 1,~m\ge 2$ is $n=8$ for which $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$.

math.NT↗

arXiv2025

On some ternary Diophantine equations of Signature $(p,p,k)$

In this paper, we summarize the work on ternary Diophantine equation of the form $Ax^n+By^n=cz^m$, where $m \in \{2,3,n\} $, $n\geqslant 7 $ is a prime. Moreover, we completely solve some particular cases ($A=5^α, ~B=64,~ c=3, ~m=2; \quad A=2^α,~ B=27, ~c \in \{7,13\}, ~m=3$).

math.NT↗

arXiv2025

Quantitative growth of linear recurrences

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)α_i^n.$ Assume $\max_i \vert α_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that for any $ε>0$ there is a effectively computable constant $C(ε),$ such that if $ \vert u_n \vert < (\max_i\{ \vert α_i \vert \})^{n(1-ε)}$, then $n<C(ε)$. Using results of Schmidt and Evertse, a complete non-effective (qualitative) proof of this conjecture was given by Fuchs and Heintze (2021) and, independently, by Karimov and al.~(2023). In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max_i\{ \vert α_i \vert \})^{n(1-ε)}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

math.NT↗

arXiv2022

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003).

math.NT↗