Related papers: Lucas sequences whose 8th term is a square

Let Q(N;q,a) denotes the number of squares in the arithmetic progression qn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over all non-trivial arithmetic progressions qn + a. Rudin's conjecture asserts that Q(N)=O(Sqrt(N)),…

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we get the explicit expressions of all squares, and then establish the tree structure of the positions of repeated squares…

For every positive integer $n$ greater than $4$ there is a set of Latin squares of order $n$ such that every permutation of the numbers $1,\ldots,n$ appears exactly once as a row, a column, a reverse row or a reverse column of one of the…

The Gauss--Lucas theorem states that any convex set $K\subset\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\in\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following…

We define a countable family of sequences of points in the unit square: the {\it $LS$-sequences of points \`a la Halton}. They reveal a very strange and interesting behaviour, as well as resonance phenomena, for which we have not found an…

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of…

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the…

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every…

Let \(\mathcal{P}(n)\) be the set of partitions of the positive integer \(n\). For \(\alpha=(\alpha_1,...,\alpha_t) \in \mathcal{P}(n)\) define the diagonal sequence \(\delta(\alpha)=(d_k(\alpha))_{k \geq 1}\) via \( d_k(\alpha) =…

We find all elliptic curves defined over $\mathbb{Q}$ that have a rational point of order $N$, $N\ge 4$, and whose conductor is of the form $p^aq^b$, where p, q are two distinct primes, a, b are two positive integers. In particular, we…

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of…

For integers $k \geq 2$, the $k$-generalized Lucas sequence $\{L_n^{(k)}\}_{n \geq 2-k}$ is defined by the recurrence relation \[ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \quad \text{for } n \geq 2, \] with initial terms given by…

We construct quadratic finite-dimensional Poisson algebras and their quantum versions related to rank N and degree one vector bundles over elliptic curves with n marked points. The algebras are parameterized by the moduli of curves. For N=2…

Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$…

We study universal cycles of the set ${\cal P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with ${\cal P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or…

We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant.…

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of the $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…