arXiv subjects

Eric F. Bravo

Publications and source records attributed to Eric F. Bravo.

4 recordsLinked to original sources

arXiv2026

On Brocard's problem with Padovan and Perrin numbers

The Padovan sequence $\{P_{m}\}_{m\ge 0}$ is a ternary recurrence sequence with companion polynomial $X^{3}-X-1$ and initial conditions $P_{0}=P_{1}=P_{2}=1$. The Perrin sequence $\{R_{m}\}_{m\ge 0}$ is defined by the same companion polynomial as the Padovan sequence, but has initial values $R_{0}=3$, $R_{1}=0$, and $R_{2}=2$. We solve the Brocard-Ramanujan equation $n!+1=x^{2}$, where $n!$ is the factorial of $n$ and $x$ is a Padovan number or a Perrin number. In both cases, we prove that $(n,x)=(4,5)$ is the only solution.

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↗

arXiv2020

$k$--Fibonacci numbers with two blocks of repdigits

A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we find all $k$--Fibonacci numbers that are concatenations of two repdigits. This generalizes prior results which dealt with the above problem for the particular cases of Fibonacci and Tribonacci numbers.

math.NT↗

arXiv2014

Coincidences in generalized Lucas sequences

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers that appear in different generalized Lucas sequences; i.e., we study the Diophantine equation $L_n^{(k)}=L_m^{(\ell)}$ in nonnegative integers $n,k,m,\ell$ with $k, \ell\geq 2$. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is a continuation of the earlier work [4].

math.NT↗