Related papers: Almost Repdigits in $ k-$generalized Lucas Sequenc…

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = \alpha_i\alpha_{i-1}\dots \alpha_0$ and $B = \beta_i\beta_{i-1}\dots \beta_0$. Lucas proved that $$\binom{A}{B} \equiv…

Let $k \geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\mathcal{P}(n) = \prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $\Omega (\mathcal{P} (n) ) \leq \rho_k$ infinitely…

A \emph{repdigit} is a natural number greater than 10 which has all of its base-10 digits the same. In this paper we find all examples of two repdigits adding to a square. The proofs lead to interesting questions about consecutive quadratic…

In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that…

In this paper, by presenting bi-periodic Lucas numbers as a binomial sum, we introduce the bi-periodic incomplete Lucas numbers. After that, by using the bi-periodic incomplete Lucas numbers, we derive the recurrence relation and the…

Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…

We consider numbers formed by concatenating some of the base b digits from additive functions f(n) that closely resemble the prime counting function \Omega(n). If we concatenate the last \lceil y \frac{\log \log \log n}{\log b} \rceil…

Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $|L(\tfrac12,\chi)|^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q)\ll_k \phi(q)(\log q)^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$…

Let $T_{k}$ be the $k^{\textrm{th}}$ Tribonacci number and $L_{n}$ be the $n^{\textrm{th}}$ Lucas number defined by their respective recurrence relation $T_{k}=T_{k-1}+T_{k-2}+T_{k-3}$ and $L_{n}=L_{n-1}+L_{n-2}$. In this study, we solve…

In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients…

Let $k\ge 2$ and $\{L_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$-Lucas numbers whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we solve the Diophantine equation…

Let $k \in \mathbb{N}$ and suppose we are given $k$ integers $1 \leq a_1, \dots, a_k \leq n$. If $\sqrt{a_1} + \dots + \sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\gtrsim…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…

The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been…

In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the…

(Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. Historically, there has been much interest in when the…

For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas…