We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and $u_{n+2}(a)=au_{n+1}(a)-u_{n}(a)$ for $n \geq 0$. We show that for a given $a \geq 3$, there is at most one $n \geq 5$ such that $u_{n}(a)$ is a near-square. With the exceptions of $u_{6}(3)=12^{2}$ and $u_{7}(6)=239 \cdot 13^{2}$, any such $u_{n}(a)$ can only be a near-square if $a \equiv 2 \bmod 4$, $n \equiv 3 \bmod 4$ is prime and $n \geq 19$.
This is part of a more general phenomenon regarding near-squares in non-degenerate recurrence sequences defined for integers $a$ and $b=-b_{1}^{2}$ by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$ and $u_{n+2}(a,b)=au_{n+1}(a,b)+bu_{n}(a,b)$ for $n \geq 0$ (see our Conjecture 1.1). It arises from a new Aurifeuillean-like factorization of elements of recurrence sequences that we have discovered (see relation (1.1)).
Let $ \prod_{i=1}^d (X-α_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $ε_1,\dots,ε_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-α_i ε_i^aY)$, $a\in {\mathbb Z}$, which we write as $$ X^d-U_1(a)X^{d-1}Y+\cdots+(-1)^{d-1} U_{d-1}(a) XY^{d-1}+(-1)^d U_d(a) Y^d.$$ In this paper we study these sequences $\bigl(U_h(a)\bigr)_{a\in {\mathbb Z}}$ which turn out to be linear recurrence sequences.
In this paper, we present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P,Q,R_0$, and $R_1$ be fixed integers and let $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence sequence defined by $R_{n+2}=PR_{n+1}-QR_{n}$ $(\forall n\geq 0)$. Under some conditions on the parameters, we determine a rational nontrivial divisor for $L_{k,n}:=\mathrm{lcm}\left(R_k,R_{k+1},\dots,R_n\right)$, for all positive integers $n$ and $k$, such that $n\geq k$. As consequences, we derive nontrivial effective lower bounds for $L_{k,n}$ and we estab…
Given a sequence of distinct positive integers $w_0 , w_1, w_2, \ldots$ and any positive integer $n$, we define the discriminator function $\mathcal{D}_{\bf w}(n)$ to be the smallest positive integer $m$ such that $w_0,\ldots, w_{n-1}$ are pairwise incongruent modulo $m$. In this paper, we classify all binary recurrent sequences $\{w_n\}_{n\geq 0}$ consisting of different integer terms such that $\mathcal{D}_{\bf w}(2^e)=2^e$ for every $e\geq 1.$ For all of these sequences it is expected that one can actually give a fairly simple description of $\mathcal{D}_{\bf w}(n)$ for every $n\ge 1.$ For …
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 generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence $(G_n)_n$, under weak assumptions, and a given polynomial $T(x)\in \mathbb{Z}[x]$, we shall prove that if $G_n=mx^m+T(x)$, then \[ m\ll\log \log |x|\log^2(\log \log |x|)\ \mbox{and}\ n\ll\log |x|\l…
We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It provides several integer sequences, whose higher-order homogeneous recurrences are determined by the help of a special matrix recurrence.
For every positive integer $n$ and every $δ\in [0,1]$, let $B(n, δ)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $δ$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $α$ and $β$, with $|α| \geq |β|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $α/ β$ is not a root of …
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of the type $\sum_{m=0}^n B_{k m+r}B_{k(n-m)+r}$ ($k>r\ge 0$) and the alternating sum of reciprocal of balancing numbers $\left\lfloor\left(\sum_{k=n}^\infty\frac{1}{B_{l k}}\right)^{-1}\right\rfloor$. Similar results are also obtained for Lucas-balancing numbers $C_n$ ($n=0,1,\cdots$), satisfying the binary recurrence $C_n=6C_{n-1}-C_{n-2}$ ($n\ge 2$) with $C_0=…
In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group $\mathbb{Z}/p\mathbb{Z}^\times$ for infinitely many $p$. In \cite{MoSt}, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture (\cite{Ho}). In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the f…
We show how to prove theorems in additive number theory using a decision procedure based on finite automata. Among other things, we obtain the following analogue of Lagrange's theorem: every natural number > 686 is the sum of at most 4 natural numbers whose canonical base-2 representation is a binary square, that is, a string of the form xx for some block of bits x. Here the number 4 is optimal. While we cannot embed this theorem itself in a decidable theory, we show that stronger lemmas that imply that the theorem can be embedded in decidable theories, and show how automated methods can be us…
Zeckendorf proved that every positive integer can be expressed as the sum of non-consecutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game. Two players begin with $n \ 1$'s and take turns applying rules inspired by the Fibonacci recurrence, $F_{n+1} = F_n + F_{n-1}$, until a decomposition without consecutive terms is reached; whoever makes the last move wins. We look at a game resulting from a generalization of the Fibonacci numbers, the Fibonacci Quilt sequence. These arise from the two-dimensional geometric property of tiling the plane through the Fibonacci s…
