What is the period of the Fibonacci sequence modulo a prime? The purpose of our brief expository paper is to illustrate an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra (rather than the case-by-case analysis found in many papers on the subject, or techniques from graduate number theory). Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in undergraduate course in abstract algebra or Galois theory.
We study the subgraph of the Young-Fibonacci graph induced by elements with odd $f$-statistic (the $f$-statistic of an element $w$ of a differential graded poset is the number of saturated chains from the minimal element of the poset to $w$). We show that this subgraph is a binary tree. Moreover, the odd residues of the $f$-statistics in a row of this tree equidistibute modulo any power two. This is equivalent to a purely number theoretic result about the equidistribution of residues modulo powers of two among the products of distinct odd numbers less than a fixed number.
We show that for primes $p < 10^{14}$ the period length $κ(p^2)$ of the Fibonacci sequence modulo $p^2$ is never equal to its period length modulo $p$. The investigation involves an extensive search by computer. As an application, we establish the general formula $κ(p^n) = κ(p) \cdot p^{n-1}$ for all primes less than $10^{14}$.
Let $p$ be an odd prime. It is well known that $F_{p-(\frac p5)}\equiv 0\pmod{p}$, where $\{F_n\}_{n\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\not=5$ then we may determine $F_{p-(\frac p5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k}{(-16)^k}\equiv\left(\frac{p}5\right)\left(1+\frac{F_{p-(\frac {p}5)}}2\right)\pmod{p^3}.$$ We also use Lucas quotients to determine $\sum_{k=0}^{(p-1)/2}\binom{2k}k/m^k$ modulo $p^2$ for any integer $m\not\equiv0\pmod{p}$; in particular, we obtain $$\sum_{k=0}^{(p-1)/2}\frac{\bino…
Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of $2$ modulo $F_n$, for every positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use of the so-called base-$φ$ expansion of real numbers.
In this paper, we study natural numbers $m$ with ${u_{z \pm 1} \equiv -1 \ (m)}$ for a $z \in \mathbb{N}$, where $u_n$ is the $n$th Fibonacci number. Furthermore, we want to show for $m \in \mathbb{N}\setminus\{0,1\}$: $[\forall p \ \text{prim}: p^2 \nmid u_{γ(p)}] \Leftrightarrow [m^2 \mid u_{γ(m)} \Rightarrow m \in \{6,12\}]$.
We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.
By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that it yields a transcendental element in the $\mathbb{Q}$-algebra $\mathscr{A}$ of integers modulo infinitely large primes under the generalized Riemann hypothesis.
In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where $p$ is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, $m$ and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.
The generalized Fibonacci sequences are sequences $\{f_n\}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t \in \mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the $p$-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo $n$ in more d…
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 $α(p) + 1$, where $α(m)$ is the restricted period length of the Fibonacci sequence modulo $m$.
We prove recurrence relations and modulo periodic properties of multiple derivatives of Fibonacci polynomials. We apply the obtained results to present the dynamic structures of Fibonacci polynomials over the ring of 2-adic integers by investigating minimal decompositions which consist of minimal subsystems and attracting basins.
In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequence of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute them, and we also obtain some properties about the quotient of the period and the rank. We find that the polynomial case is much more complicated than the integer case.
Let $\{G_n\}$ be a periodic sequence of integers modulo $m$ and let $\{SG_n\}$ be the partial sum sequence defined by $SG_n:= \sum_{k=0}^nG_k $ (mod $m$). We give a formula for the period of $\{SG_n\}$. We also show that for a generalized Fibonacci sequence $F(a,b)_n$ such that $F(a,b)_0=a$ and $F(a,b)_1=b$, we have $$S^i F(a,b)_n = S^{i-1}F(a,b)_{n+2}-{n+i \choose i-2}a-{n+i \choose i-1} b$$ where $S^i F(a,b)_n $ is the i-th partial sum sequence successively defined by $S^i F(a,b)_n := \sum_{k=0}^n S^{i-1}F(a,b)_k$. This is a generalized version of the well-known formula $$\sum_{k=0}^n F_k = …
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q…
In this paper, we prove a few lemmas concerning Fibonacci numbers modulo primes and provide a few statements that are equivalent to Wall-Sun-Sun Prime Conjecture. Further, we investigate the conjecture through heuristic arguments and propose a few additional conjectures for future research.
For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.
Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Ch…
In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of such a division field. Such fields include a large class of non-solvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the elliptic curve reduced modulo p.
Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\{f_n\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be related to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period a…
Second order Recurrences, quadratic number fields and cyclic codes
Anhui University（安徽大学）；École Supérieure de l’Education et de la Formation（教育与高等师范学院）；Sogang University（西江大学）；Aix Marseille Univ, CNRS（艾克斯-马赛大学，法国国家科学研究中心）
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).
Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\{f_n\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be rela ted to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period …
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffice…
We prove that the number of unimodular integral matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field $F$. Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes de…
We study integral models, so-called Pappas-Rapoport or splitting models, of some PEL Shimura Varieties whose data are ramified at a prime p. We show that except in a specific case, these models are smooth when there is no level at p, and we study their special fiber modulo p. We show that interesting stratifications appear, and we show that the $μ$-ordinary locus is open and dense, following a strategy of Wedhorn in the unramified case.
