Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

一些包含Fibonacci-Lucas数的恒等式和同余式

给出了一些包含F ibonacci-Lucas数的恒等式和同余式.     

关于Lucas三角形猜想的一个结果

运用初等方法,证明k=7时Lucas三角形不存在.     

关于Fibonacci三角形猜想k=6的证明

运用初等方法,证明k=6时Fibonacci三角形不存在.     

A New Approach to Polynomial Identities

Using the formal derivative idea, we give a generalization for the Cauchys Theorem relating to the factors of (x + y)ⁿ–xⁿ– yⁿ. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial

Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials

In this article, we find elements of the Lucas polynomials by using two matrices. We extend the study to the n-step Lucas polynomials. Then the Lucas polynomials and their relationship are generalized in the paper. Furthermore, we give relationships between the Fibonacci polynomials and the Lucas polynomials.     

The Lucas p-matrix

In this note, we study the Fibonacci and Lucas p-numbers. We introduce the Lucas p-matrix and companion matrices for the sums of the Fibonacci and Lucas p-numbers to derive some interesting identities of the Fibonacci and Lucas p-numbers.     

Some new Fibonacci and Lucas identities by matrix methods

The aim of this article is to characterize the 2 × 2 matrices X satisfying X ² = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers.     

Maximal and minimal congruences on some semigroups

In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup Zn under multiplication modulo n. And, when Y lohtain in X, we do the same for the semigroup T（X, Y） consisting of all elements of T（X） whose range is contained in Y. We also characterise the minimal congruences on T（X. Y）.     

关于一些级数取整值问题的注记

利用Pell方程的理论,讨论了与广义Lucas数有关的一些级数的取整值问题.在一定条件下,解决了一些级数的取整值问题.     

k=2t·3时不存在以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.利用平方剩余的方法得到:当k=2'·3时,不存在边长为(Fn-k,Fn,Fn)的Fibonacci三角形(k＜2).     

探索以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.存在以(F<,n-k>,F<,n>.F<,n>)为边长的Fibonacci三角形的情形可以被划分为三类(k时,不存在边长为(F<,n-k>,F<,n>.F<,n>)的Fibonacci三角形.     

线性递归序列的矩阵表示

先给出k阶线性递归序列的矩阵表示,然后用矩阵方法得到2阶线性递归序列的通项及一些恒等式.     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

关于与广义Fibonacci,Lucas数有关的一些级数的注记

求出了一些与广义Fibonacci,Lucas数有关的一些倒数级数的值。     

涉及广义Fibonacci-Lucas数的幂的一些级数的近似值

计算出涉及广义Fibonacci-Lucas数的幂的一些级数的近似值。     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

关于Lucas数列同余性质的研究

将二项式系数的性质应用到Lucas数列的研究中,并结合Fibonacci数列与Lucas数列的恒等式得到几个有趣的Lucas数列的同余式.     

New Fibonacci and Lucas primes

Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers have been determined for and all probable prime Lucas numbers have been determined for . A rigorous proof of primality is given for and for numbers with , , , , , , , , the prime having 3020 digits. Primitive parts and of composite numbers and have also been tested for probable primality. Actual primality has been established for many of them, including 22 with more than 1000 digits. In a Supplement to the paper, factorizations of numbers and are given for as far as they have been completed, adding information to existing factor tables covering .

     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.