Related papers: Fibonacci Sequences And Real Quadratic p-Rational …
Let $p$ be an odd prime, and $m,r \in \mathbb{Z}^+$ with $m$ coprime to $p$. In this paper we investigate the real quadratic fields $K = \mathbb{Q}(\sqrt{m^2p^{2r} + 1})$. We first show that for $m < C$, where constant $C$ depends on $p$,…
We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…
Choose a random degree d poly f with coefficients in a finite field F. We estimate the ultimate period of f under compositional iteration. We also determine the joint distribution of the small cycle lengths in the graph with edges (x,f(x)),…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in…
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper…
Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF…
The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $\Gamma^{(p)}_n$, which are subgraphs of $Q_n$ induced by…
This paper proposes a computational method for obtaining the length of the cycle that arises from the Fibonacci series taken mod m (some number) and mod p (some prime number).
Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…
We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.
In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general…
Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at…
A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are…
In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.
We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.
A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period…
In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the…
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…