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).

Markoff–Rosenberger triples and generalized Lucas sequences

We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ a x 2 + b y 2 + c z 2 = d x y z with $$(x,y,z)=(U_i,U_j,U_k)$$ ( x , y , z ) = ( U i , U j , U k ) , where $$U_i$$ U i denotes the i-th generalized Lucas number of first/second kind. We provide an upper bound for the minimum of the indices and we apply the result to completely resolve concrete equations, e.g. we determine solutions containing only balancing numbers and Jacobsthal numbers, respectively.

MIN MATRICES WITH HYPER LUCAS NUMBERS

In this paper, we examine Min matrix L[Lk+min(i,j)-1]i,j=1 where Ln(r) denotes the nth hyper-Lucas number of order r. We mainly focus on characteristic polynomial of L. Also, we compute determinants, inverses of L and its Hadamard inverse. Moreover, we give a numerical example to illustrate our results.

Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n × n Foeplitz matrix is sparse and can be expressed by the nth and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n × n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n × n ( n > 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.

Lucas numbers of the form (2t/k)

Let Lm denote the mth Lucas number. We show that the solutions to the diophantine equation (2t/k) = Lm, in non-negative integers t, k ≤ 2t−1, and m, are (t, k, m) = (1, 1, 0), (2, 1, 3), and (a, 0, 1) with non-negative integers a.