Repdigits as difference of two Fibonacci or Lucas numbers

In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)\in\{(6,4),(7,4),(7,6),(8,2)\},$ where $n>m.$Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},\33=F_{9}-F_{1}=F_{9}-F_{2},\55=F_{11}-F_{9}=F_{12}-F_{11},\88=F_{11}-F_{1}=F_{11}-F_{2},\555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6},\ 22=L_{7}-L_{4},\4=L_{8}-L_{2}$ (Theorem 3).

An elementary unified approach to prove some identities involving Fibonacci and Lucas numbers

Using the explicit formulas of the generating polynomials of Fibonacci and Lucas, we prove some new identities involving Fibonacci and Lucas numbers. As an application of these identities, we show how some Diophantine equations have infinitely many solutions. To illustrate the powerful of this elementary method, we give proofs of many known formulas.

A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.