Related papers: Pell and Pell-Lucas numbers as difference of two r…

In this paper, we find all repdigits which can be expressed as the product of a Narayana, and a product of two repdigits is Narayana.

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers $ d $ such that the Pell…

For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial…

We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence,…

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…

The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with…

Pandigital and penholodigital numbers are numbers that contain every digit or nonzero digit respectively. We study properties of pandigital or penholodigital numbers that are also square, oblong or prime.

A repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number has the form $d(10^m-1)/9$ for some $m\geq 1$ and $1 \leq d \leq 9$. Let $\left(T_n\right)_{n\ge0}$ be the sequence of Tribonacci.…

By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map…

The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…

A Cullen number is a number of the form $m2^m+1$, where $m$ is a positive integer. In 2004, Luca and St\u anic\u a proved, among other things, that the largest Fibonacci number in the Cullen sequence is $F_4=3$. Actually, they searched for…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking…

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} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for…

The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2,…

We study multiplicative dependence between terms of the $k$-generalized Pell sequence $(P_n^{(k)})_{n\ge 2-k}$, defined by the linear recurrence \[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, \] with initial…

We show that if $\{U_n\}_{n\geq 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=C_{m_1}C_{m_2}\cdots C_{m_k}$ with $1\leq m_1\leq m_2\leq \cdots\leq m_k$, where $C_m$ is the $m$th Catalan number satisfies $n<6500$. In case…

A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms.…

For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the…

Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two…