The Period and the Distribution of the Fibonacci-like Sequence Under Various Moduli

We reduce the Fibonacci sequence mod m for a natural number m, and denote it by F (mod m ). We are going to introduce the properties of the period and distribution of F (mod m). That is, how frequently each residue is expected to appear within a single period. These are well known themes of the research of the Fibonacci sequence, and many remarkable facts have been discovered. After that we are going to study the properties of period and distribution of a Fibonacci-like sequence that the authors introduced in article in the previous issue of Undergraduate Math Journal. This Fibonacci-like sequence also has many interesting properties, and the authors could prove an interesting theorem in this article. Some of properties are very difficult to prove, and hence we are going to present some predictions and calculations by computers

Sobre problemas envolvendo números de k-bonacci e coeficientes fibonomiais

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Os números de Fibonacci possui várias generalizações, entre elas temos a sequência (Fn (k))n que é chamada de sequência de Fibonacci k-generalizada. Observando a identidade F2 n+F2 n+1=F2n+1, Chaves e Marques, em 2014, provaram que a equação Diofantina (Fn (k))2+ (F(k) n+1)2= Fm (k) não possui soluções em inteiros positivos n, m e k, com n > 1 e k ≥ 3. Nesse trabalho, mostramos que a equação Diofantina (Fn (k))2 +(F(k) n+1)2 = Fm (l), não possui solução para 2≤ k k + 1. Outra generalização da sequência de Fibonacci s˜ao os coeficientes fibonomiais. Em 2015, Marques e Trojovský provaram que uma condição mais fraca. se p ≡ ± 1 (mod 5), então p † [pa+1 pa] , para todo a ≥ 1.Nesse trabalho, encontramos as classe de resíduos de módulo p, p2, p3 e p4, quando p ≡ ± 1 (mod 5) e sobre uma condição mais fraca. Em particular, provamos que se p é um número primo tal que p ≡ ± 1 (mod 5), então [pa+1 pa] ≡ 1 (mod p).Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).Regarding the identity F2 n+F2 n+1=F2n+1, Chaves and Marques, in 2014, proved that (Fn (k))2+ (F(k) n+1)2= Fm (k) does not have solution for integers n, m e k, with n > 1 and k ≥ 3. In this work, we show that (Fn (k))2 +(F(k) n+1)2 = Fm (l) does not have solutions for 2≤ k k + 1. Another generalization of the Fibonacci sequence are the Fibonomial coe#cients. In 2015, Marques and Trojovský proved that if p ≡ ± 1 (mod 5), then p † [pa+1 pa] for all a ≥ 1. In this work, we also find the residue class of [pa+1 pa] modulo p, p2, p3 e p4, when p ≡ ± 1 (mod 5) under some weak hypothesis. In particular, we proved that if p is a prime number such that p ≡ ± 1 (mod 5), then [pa+1 pa] ≡ 1 (mod p)

pp-regularity of the pp-adic valuation of the Fibonacci sequence

We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $\alpha(p) + 1$, where $\alpha(m)$ is the restricted period length of the Fibonacci sequence modulo $m$.Comment: 7 pages; publication versio