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Prosper Kaggwa

Publications and source records attributed to Prosper Kaggwa.

6 recordsLinked to original sources
arXiv2026

On Wagstaff primes in the $k$-Pell number sequence

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. For an integer $k\ge 2$, the sequence of $k$-generalized Pell numbers $\bigl(P_n^{(k)}\bigr)_{n\ge 2-k}$ is the linear recurrence of order $k$ given by $P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}$ for $n\ge 2$, with initial values $P_{2-k}^{(k)}=\cdots=P_{0}^{(k)}=0$ and $P_1^{(k)}=1$. For $k=2$ this is the classical Pell sequence. Motivated by the recent determination of the Wagstaff primes in the $k$-Fibonacci and $k$-Lucas sequences, we study the exponential Diophantine equation $P_n^{(k)}=(2^{\mathfrak{p}}+1)/3$ in positive integers $n,k$ and an odd prime $\mathfrak{p}\geq 3$, with $k\ge 2$. We show that this equation has no solutions, that is no $k$-Pell number is a Wagstaff prime. The proof combines linear forms in logarithms of algebraic numbers with reduction based on the theory of continued fractions, together with a computational verification in the remaining finite range.

math.NT↗
arXiv2026

$k$-Fibonacci and $k$-Lucas numbers with the Hölder inequality

Dujella, Jak\v setić and Pe\v carić recently established in \cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities $\sum_{i=1}^nF_i^2=F_nF_{n+1}$ and $\sum_{i=1}^nF_iF_{i+1}=F_{n+1}^2-\frac{1+(-1)^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \begin{align*} \sum_{i=1}^nL_{k,i}^2=\frac{L_{k,n}L_{k,n+1}-2k}{k},\qquad \sum_{i=1}^nF_{k,i}^2=\frac{F_{k,n}F_{k,n+1}}{k}, \end{align*} and \begin{align*} \sum_{i=1}^nL_{k,i}L_{k,i+1}=\frac{L_{k,n+1}^2}{k}-k+\bigl((-1)^n-1\bigr)\left(\frac{2}{k}+\frac{k}{2}\right), \end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}L_{k,i}=F_{k,2i}$, valid for every $k\ge1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.

math.NT↗
arXiv2026

On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas number. Mathematically, we study of two Diophantine equations $L_n=b^dL_m+F_k$ and $L_n=b^dF_m+L_k$, where $d$ is the number of digits of $F_k$ or $L_k$ in base $b$. To tackle these equations, we combine tools from Diophantine approximation on non-zero linear forms in logarithms and reduction methods based on continued fractions. This allows us to prove that only finitely many such Lucas numbers exist.

math.NT↗
arXiv2025

On concatenations of two $k$-generalized Lucas numbers

For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2 \), \( L_1^{(k)} = 1 \) for all \( k \geq 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \geq 3 \). In this paper, we determine all \( k \)-generalized Lucas numbers that are concatenations of two terms of the same sequence and completely solve this problem for \( k \geq 3 \). Our approach combines nonzero lower bounds for linear forms in logarithms, reduction techniques based on the Baker--Davenport method and the LLL-algorithm, together with continued fraction analysis and computational verification using SageMath.

math.NT↗
arXiv2025

On Palindromic forms in the $k$-Lucas sequence composed of two distinct Repdigits

For integers $k \geq 2$, the $k$-generalized Lucas sequence $\{L_n^{(k)}\}_{n \geq 2-k}$ is defined by the recurrence relation \[ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \quad \text{for } n \geq 2, \] with initial terms given by $L_0^{(k)} = 2$, $L_1^{(k)} = 1$, and $L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0$. In this paper, we extend work in \cite{Lucas} and show that the result in \cite{Lucas} still holds for $k\ge 3$, that is, we show that for $k\ge 3$, there is no $k$-generalized Lucas number appearing as a palindrome formed by concatenating two distinct repdigits.

math.GM↗