arXiv subjects

Kouèssi Norbert Adédji

Publications and source records attributed to Kouèssi Norbert Adédji.

6 recordsLinked to original sources

arXiv2026

On the Diophantine Equations Arising from Three-Term $b$-Concatenations in Pell-Type Sequences

We study the Diophantine problem of representing a Pell or Pell--Lucas number as the base-$b$ concatenation of three other Pell or Pell--Lucas numbers. Under the condition that the index of the middle block does not exceed that of the leading block, we obtain an explicit upper bound for the sequence indices in terms of $b$. For $2\leq b\leq15$, we determine exactly $51$ solutions up to the identification $Q_0=Q_1=2$, while there are $74$ solutions when distinct index choices are counted separately.

math.NT↗

arXiv2026

Sums of three Fibonacci numbers as concatenations of three repdigits in base $b$

In this paper, we investigate sums of three Fibonacci numbers that can be expressed as concatenations of three repdigits in base $b$, where $b\ge 2$ is an integer. We prove that for bases $2\le b\le 10$, only finitely many such sums exist, and we determine all of them explicitly. Among these solutions, the largest occurs for $b=4$ and is given by $$ F_{42}+F_{29}+F_{20}=268435290=\overline{33333333331122}_4. $$

math.GM↗

arXiv2026

Thabit and Williams Numbers Base $b$ as a Sum or Difference of Two $g$-Repdigits

We investigate cases where Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$-repdigits. For specific values of $b$ and $g$, we describe parametric solutions yielding infinitely many solutions for some equations and establish upper bounds for the parameters of the remaining finitely many solutions. As an illustration, we also provide a complete solution for some equations.

math.NT↗

arXiv2024

Narayana's cows numbers which are concatenations of three repdigits in base $ρ$

Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $ρ\geqslant2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number $ \mathcal{N}_n$ which is concatenations of three repdigits in base $ρ$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7ρ$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $ρ$ with $1 \leqslant ρ\leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7.$

math.NT↗

arXiv2024

On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers

Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and $P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m, n, k, b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le 10.$

math.NT↗

arXiv2022

On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$

Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular.

math.NT↗