Searcharxiv⌕ Search

arXiv subjects

J. C. Saunders

Publications and source records attributed to J. C. Saunders.

16 recordsLinked to original sources
arXiv2025

Residues of Terms of Lucas Sequences Modulo $3^k$

The Fibonacci sequence defined by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ has a shortest period length of $4\cdot 3^{k-1}$ modulo $3^k$ for every $k\in\mathbb{N}$. In 2011, Bundschuh and Bundschuh \cite{bundschuh3} gave the frequencies of every residue $0\leq b\leq 3^k-1$ in this shortest period. In particular, their result implies that the Fibonacci sequences is not stable modulo $3$. Here we extend this result to other Lucas sequences. More specifically, we give analogous results for Lucas sequences defined by $\left(u_n\right)_n$ with $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, as well as Lucas sequences defined by $\left(v_n\right)_n$ with $v_0=2$, $v_1=P$, and $v_n=Pv_{n-1}+v_{n-2}$ for all $n\geq 2$. In particular, our result implies that none of these Lucas sequences are stable modulo $3$ either.

math.NT↗
arXiv2025

Products of Tribonacci Numbers that are the Products of Factorials

In 2014 Marques and Lengyel gave all of the solutions to the equation $T_n=m!$, where $T_n$ is the $n$th term of the Tribonacci sequence $0,1,1,2,4,7,13,24,\ldots$. In 2023 Alahmadi and Luca generalized their result to the equation $T_n=m_1!m_2!\cdots m_k!$ for every $k\in\mathbb{N}$, where $m_1\leq m_2\leq\ldots\leq m_k$ listing all the solutions to this equation. Here we generalize these results further and give all the solutions to $T_nT_{n+1}T_{n+2}\cdots T_{n+r}=m_1!m_2!\cdots m_k!$ and $ |T_{-n}T_{-n-1}T_{-n-2}\cdots T_{-n-r}|=m_1!m_2!\cdots m_k!$ for every $n,r\in\mathbb{N}$, where $m_1\leq m_2\leq\ldots\leq m_k$.

math.NT↗
arXiv2024

Runs of integers with constant values of the Carmichael function

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $λ(n+1)=λ(n+2)=\cdots=λ(n+k)$ holds for some $n\leq x$, where $λ$ is the Carmichael $λ$ function, is bounded above by $O\left((\log x\log\log x)^2\right)$. Their method involved bounding the value of $λ(n + i)$ from below using the prime factorization of $n + i$ for each $i \leq k$. They then used the fact that every $λ(n + i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime.

math.NT↗
arXiv2022

Primes and composites in the determinant Hosoya triangle

In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{determinant Hosoya triangle} which we denote by ${\mathcal H}$. We discuss the divisibility properties of the above numbers and their primality. We give a small sieve of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. We prove that ${\mathcal H}$ has arbitrarily large neighbourhoods of composite entries. Finally we present an abundance of data indicating a very high density of primes in ${\mathcal H}$.

math.NT↗
arXiv2022

The number of k-tons in the coupon collector problem

Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ and that we keep buying boxes until we collect at least $m$ coupons of each type. For $k\geq m$ call a certain coupon a $k$-ton if we see it $k$ times by the time we have seen $m$ copies of all of the coupons. Here we determine the asymptotic distribution of the number of $k$-tons after we have collected $m$ copies of each coupon for any $k$ in a restricted range, given any fixed $m$. We also determine the asymptotic joint probability distribution over such values of $k$ and the total number of coupons collected.

math.PR↗
arXiv2022

Endomorphisms of positive characteristic tori: entropy and zeta function

Let $F$ be a finite field of order $q$ and characteristic $p$. Let $\mathbb{Z}_F=F[t]$, $\mathbb{Q}_F=F(t)$, $\mathbb{R}_F=F((1/t))$ equipped with the discrete valuation for which $1/t$ is a uniformizer, and let $\mathbb{T}_F=\mathbb{R}_F/\mathbb{Z}_F$ which has the structure of a compact abelian group. Let $d$ be a positive integer and let $A$ be a $d\times d$-matrix with entries in $\mathbb{Z}_F$ and non-zero determinant. The multiplication-by-$A$ map is a surjective endomorphism on $\mathbb{T}_F^d$. First, we compute the entropy of this endomorphism; the result and arguments are analogous to those for the classical case $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Second and most importantly, we resolve the algebraicity problem for the Artin-Mazur zeta function of all such endomorphisms. As a consequence of our main result, we provide a complete characterization and an explicit formula related to the entropy when the zeta function is algebraic.

math.NT↗
arXiv2022

A general criterion for the Pólya-Carlson dichotomy and application

We prove a general criterion for an irrational power series $f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n$ with coefficients in a number field $K$ to admit the unit circle as a natural boundary. As an application, let $F$ be a finite field, let $d$ be a positive integer, let $A\in M_d(F[t])$ be a $d\times d$-matrix with entries in $F[t]$, and let $ζ_A(z)$ be the Artin-Mazur zeta function associated to the multiplication-by-$A$ map on the compact abelian group $F((1/t))^d/F[t]^d$. We provide a complete characterization of when $ζ_A(z)$ is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on $\mathbb{R}^d/\mathbb{Z}^d$ in which Baake, Lau, and Paskunas prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier together with certain patching arguments involving linear recurrence sequences.

math.NT↗
arXiv2022

The Euler Totient Function on Lucas Sequences

In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are $1$ and $2$. Here we add to these two results and prove that for any fixed natural number $P\geq 3$, if we define the sequence $\left(u_n\right)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, then the only solution to the Diophantine equation $φ\left(u_n\right)=u_m$ is $φ\left(u_1\right)=φ(1)=1=u_1$.

math.NT↗
arXiv2022

A Model of Random Industrial SAT

One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT instances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a model of random industrial SAT, in which the central generalization with respect to random SAT is the additional community structure. There has been a lot of work on the satisfiability threshold of random $k$-SAT, starting with the calculation of the threshold of $2$-SAT, up to the recent result that the threshold exists for sufficiently large $k$. In this paper, we endeavor to study the satisfiability threshold for the proposed model of random industrial SAT. Our main result is that the threshold in this model tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.

cs.DS↗
arXiv2022

Irreducibility of generalized Fibonacci polynomials

A second order polynomial sequence is of Fibonacci-type $\mathcal{F}_{n}$ (Lucas-type $\mathcal{L}_{n}$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are irreducible if and only if $n$ is a prime number. For example, the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials are irreducible when $n$ is a prime number; and Chebyshev polynomials (second kind), Morgan-Voyce polynomials (Fibonacci type), and Vieta polynomials are reducible when $n$ is a prime number. In this paper we give some theorems to determine whether the Fibonacci type polynomials and Lucas type polynomials are irreducible when $n$ is prime.

math.NT↗
arXiv2021

Density of sequences of the form $x_n=f(n)^n$ in [0,1]

In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as $x_n=(\cos n)^n$ distributed themselves$\pmod 1$. Strauch's inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence $x_n=(\cos αn)^n\pmod 1$ is dense in $[0,1]$ for any fixed real number $α$ such that $α/π$ is irrational. Here we generalise Luca's results to other sequences of the form $x_n=f(n)^n\pmod 1$. We also examine the size of the set $|\{n\leq N:r<|\cos(nπα)|^n\}|$ where $0<r<1$ and $α$ are fixed such that $α/π$ is irrational.

math.NT↗
arXiv2021

Sieve Methods in Random Graph Theory

In this paper, we apply the Turan sieve and the simple sieve developed by R. Murty and the first author to study problems in random graph theory. In particular, we obtain upper and lower bounds on the probability of a graph on n vertices having diameter 2 (or diameter 3 in the case of bipartite graphs) with edge probability p(n) where the edges are chosen independently . An interesting feature revealed in these results is that the Turan sieve and the simple sieve `almost completely' complement each other. As a corollary to our result, we note that the probability of a random graph having diameter 2 approaches 1 as n approaches infinity for constant edge probability p(n)=1/2. This is an appendix of a shorter version of this paper.

math.NT↗
arXiv2021

Generalised Fibonacci sequences constructed from balancing words

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference of the two preceding terms where the pluses and minuses follow a certain pattern. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and $G_n$ is the $n$th term of the resulting generalised Fibonacci sequence, then \begin{equation*} \lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation*} exists. We extend her results to recurrences of the form $G_{m+2} = α_m G_{m+1} \pm G_{m}$ if the choices of pluses and minuses, and of the $α_m$ follow a balancing word type pattern.

math.NT↗
arXiv2018

On (a,b) Pairs in Random Fibonacci Sequences

We study the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if x is the parent of y, then y has two children, namely |x-y| and x+y. This tree was studied by Benoit Rittaud who proved that any pair of integers a,b that are coprime occur as a parent-child pair infinitely often. We extend his results by determining the probability that a random infinite walk in this tree contains exactly one pair (1,1), that being at the root of the tree. Also, we give tight upper and lower bounds on the number of occurrences of any specific coprime pair (a,b) at any given fixed depth in the tree.

math.NT↗
arXiv2018

Mahler Measure of "Almost" Reciprocal Polynomials

Here we give a lower bound of the Mahler measure on a set of polynomials that are "almost" reciprocal. Here "almost" reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern breaks down for the innermost coefficients.

math.NT↗