Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998. / A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998 / Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts).

Let $A \subseteq \Z_n$ with $|A| = k$ for some $k \in \Z^+$. We consider the metric space $(\Z_n,\delta)$ in which $\delta$ is the distance metric on $\Z_n$ defined as follows: for every $x,y \in \Z_n$, $\delta(x,y) = |x-y|_n$ where $|z|_n = \min(z,n-z)$ for $z \in \{0,\ldots,n-1\}$. We say that $A$ is \emph{Erd\H{o}s-deep} if, for every $i \in \{1,2,\dots,k-1\}$, there is a positive number $d_i$ satisfying $$|\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}|=i.$$ Erd\H{o}s-deep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k-1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$. Erd\H{o}s-deep sets have primarily been considered in metric spaces $(\Z_n,\delta)$ and $(\R^d,\norm{\cdot})$ for $d = 2$, but some exploration for $d > 2$ has been done as well. We introduce the notion of an \emph{Erd\H{o}s-deep family}. Let $\mathcal{F}=\{A_1,A_2,\dots,A_s\}$, where $A_1,\ldots, A_s \subseteq \Z_n$. Then we say $\mathcal{F}$ is Erd\H{o}s-deep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k-1\}$ there is exactly one positive number $d_i$ satisfying $$\sum_{j=1}^s |\{\{x,y\} \subseteq A_j: \delta(x,y)=d_i\}|=i,$$ and no such $d_i$ for any $i \ge k$. We provide a complete existence theorem for Erd\H{o}s-deep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}s-deep families of three arithmetic progressions. Using an identity on triangular numbers, we show a general construction for larger families whose size $s$ is the square of an integer. This construction suggests the existence of Erd\H{o}s-deep families often relies on such number-theoretic identities. We define an extremal case of the Erd\H{o}s-deep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k-1\}$; such families are called Winograd families. We conjecture that Winograd families of arithmetic progressions do not exist in the metric space $(\Z,|\cdot|)$. Erd\H{o}s-deep sets in $(\Z_n,\delta)$ correspond to a class of interesting musical rhythms. We conclude this work with a variety of musical demonstrations and original compositions using Erd\H{o}s-deep rhythm families as a creative constraint in composing multi-voiced rhythms. / Graduate

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems. / Science, Faculty of / Mathematics, Department of / Graduate

The dissertation consists of two parts. The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces. We establish a series of dichotomy-type results for the discrepancy function which state that if the L¹ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.

Tesis por compendio / [ES] La tesis "Teoría Combinatoria de Números, Recurrencia de Operadores y Dinámica Lineal" se sitúa dentro del estudio de la dinámica de operadores lineales, o Dinámica Lineal. El objetivo de este trabajo es estudiar múltiples nociones de recurrencia, que pueden presentar los sistemas dinámicos lineales, y que clasificaremos mediante la Teoría Combinatoria de Números. La Dinámica Lineal estudia las órbitas generadas por las iteraciones de una transformación lineal. Las propiedades más estudiadas en esta rama durante los últimos 30 años han sido la hiperciclicidad (existencia de órbitas densas) y el caos (con sus múltiples definiciones), siendo esta un área de investigación muy activa y obteniéndose un considerable número de resultados profundos e interesantes. Nosotros nos centraremos en la recurrencia, propiedad muy estudiada para sistemas dinámicos clásicos no lineales, pero prácticamente nueva en Dinámica Lineal pues no es hasta 2014, con el artículo de Costakis, Manoussos y Parissis titulado "Recurrent linear operators", cuando se empieza a estudiar esta noción de manera sistemática en el contexto de operadores actuando en espacios de Banach. La situación básica de la que parte nuestro estudio es la siguiente: "T : X ---> X" será un operador lineal y continuo actuando sobre un F-espacio "X" , aunque a veces necesitaremos que el espacio subyacente "X" sea un espacio de Fréchet, de Banach o de Hilbert. Dado un vector "x" y un entorno "U" de "x" estudiaremos el conjunto de retorno "N_T(x,U) = { n : T^n(x) está en U }" y dependiendo de su tamaño, observado mediante la Teoría Combinatoria de Números, diremos que el vector "x" presenta una propiedad de recurrencia u otra. La memoria de la tesis se ha realizado por compendio de artículos y consta de cuatro capítulos y un apéndice: 1. Adaptación de la "versión de autor" del artículo "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), artículo núm. 109713, 36 páginas". En este se definen por primera vez las fuertes nociones de recurrencia reiterada, U-frecuente y frecuente, y sus propiedades básicas son estudiadas. Finalmente se generaliza el estudio mediante el concepto de F-recurrencia, que se conecta con la noción de F-hiperciclicidad. 2. Adaptación al formato de la tesis de la "versión de autor" revisada del artículo "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". En este se relaciona la recurrencia de operadores con la Teoría Ergódica y los sistemas dinámicos que conservan la medida. 3. Adaptación de la "versión de autor" del preprint "Questions in linear recurrence: From the T+T-problem to lineability". Se resuelve negativamente un problema abierto de 2014: Sea "T : X ---> X" un operador recurrente. ¿Es cierto que el operador "T+T" es recurrente en "X+X"? Para resolverlo introducimos la casi-rigidez, que será, para la recurrencia, la noción análoga a la propiedad débil-mezclante (topológica) para la transitividad/hiperciclicidad; y luego construimos operadores recurrentes pero no casi-rígidos en todo espacio de Banach infinito-dimensional y separable. 4. Adaptación de la "versión de autor" revisada del preprint " Recurrent subspaces in Banach spaces". En este se estudia la propiedad de espaciabilidad (existencia de un subespacio vectorial cerrado y de dimensión infinita) para el conjunto de vectores recurrentes. - Apéndice. Para conseguir un carácter auto-contenido hemos añadido un apéndice con los resultados básicos de Teoría Combinatoria de Números que se han utilizado en los trabajos que componen la memoria. Siguiendo la normativa establecida por la Escuela de Doctorado también se incluye: - Introducción; - Discusión general de los resultados; - Conclusiones. / [CAT] La tesi "Teoria Combinatòria de Nombres, Recurrència d'Operadors i Dinàmica Lineal" se situa dins de l'estudi de la dinàmica d'operadors lineals, o simplement Dinàmica Lineal. L'objectiu d'aquest treball és estudiar múltiples nocions de recurrència, que poden presentar els sistemes dinàmics lineals, i que classificarem mitjançant la Teoria Combinatòria de Nombres. La Dinàmica Lineal estudia les òrbites generades per les iteracions d'una transformació lineal. Les propietats més estudiades en aquesta branca de les matemàtiques als darrers 30 anys han estat la hiperciclicitat (existència d'òrbites denses) i el caos (amb les seves múltiples definicions), sent aquesta una àrea de recerca molt activa i obtenint-se un considerable nombre de resultats profunds i interessants. Nosaltres ens centrarem en la recurrència, propietat molt estudiada per a sistemes dinàmics clàssics no lineals, però, pràcticament nova en Dinàmica Lineal doncs no és fins al 2014, amb l'article de Costakis, Manoussos i Parissis titulat "Recurrent linear operators", quan es comença a estudiar aquesta noció de manera sistemàtica en el context d'operadors actuant en espais de Banach. La situació bàsica de la qual parteix el nostre estudi és la següent: "T : X ---> X" serà un operador lineal i continu actuant sobre un F-espai "X", encara que de vegades necessitarem que l'espai subjacent X siga un espai de Fréchet, de Banach o de Hilbert. Llavors, donat un vector "x" i un entorn "U" de "x" estudiarem el conjunt de retorn "N_T(x,U) = { n : T^n(x) està en U }" i depenent de la seva mida, observada des del punt de vista de la Teoria Combinatòria de Nombres, direm que el vector "x" presenta una o altra propietat de recurrència. La memòria de la tesi s'ha realitzat per compendi d'articles i consta de quatre capítols i un apèndix: 1. Adaptació de la "versió d'autor" revisada de l'article "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), article núm. 109713, 36 pàgines". En aquest es defineixen per primera vegada les nocions de recurrència reiterada, U-freqüent i freqüent, i les seves propietats bàsiques són estudiades. Finalment es generalitza l'estudi mitjançant el concepte de F-recurrència, que es connecta amb la noció de F-hiperciclicitat. 2. Adaptació al format de la tesi de la "versió d'autor" revisada de l'article "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". Es relaciona la recurrència d'operadors amb la Teoria Ergòdica i els sistemes dinàmics que conserven la mesura. 3. Adaptació de la "versió d'autor" del preprint "Questions in linear recurrence: From the T+T-problem to lineability". En aquest es resol un problema obert de l'any 2014: Siga "T : X ---> X" un operador recurrent. És cert que l'operador "T+T" és recurrent en "X+X"? Per resoldre'l introduïm la quasi-rigidesa, que serà, per a la recurrència, la noció anàloga a la propietat feble-barrejant (topològica) per a la transitivitat/hiperciclicitat; i després construïm operadors recurrents però no quasi-rígids en tot espai de Banach infinit-dimensional i separable. 4. Adaptació de la "versió d'autor" del preprint "Recurrent subspaces in Banach spaces". S'inclou l'estudi de la propietat d'espaiabilitat (existència d'un subespai vectorial tancat i de dimensió infinita) per al conjunt de vectors recurrents. - Apèndix:Per aconseguir un caràcter auto-contingut hem afegit un apèndix amb resultats bàsics de Teoria Combinatòria de Nombres que es donen per suposats en els treballs que componen la memòria. Seguint la normativa establerta per l'Escola de Doctorat també s'inclou: - Introducció; - Discussió general dels resultats; - Conclusions. / [EN] The thesis "Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics" is part of the study of the dynamics of linear operators, simply called Linear Dynamics. The objective of this work is to study multiple notions of recurrence, that linear dynamical systems can present, and which will be classified through Combinatorial Number Theory. Linear Dynamics studies the orbits generated by the iterations of a linear transformation. The two most studied properties in this branch of mathematics during the last 30 years have been hypercyclicity (existence of dense orbits) and chaos (with its multiple definitions), being this a very active research area with a considerable number of exceptionally deep but also interesting results. We will focus on recurrence, a property widely studied in the classical setting of non-linear dynamical systems, but practically new with respect to Linear Dynamics since it was not until 2014, with the article by Costakis, Manoussos and Parissis entitled "Recurrent linear operators", when this notion started to be systematically studied in the context of operators acting on Banach spaces. The basic situation from which our study starts is the following: "T : X ---> X" will be a continuous linear operator acting on an F-space "X", although sometimes we will need the underlying space X to be a Fréchet, Banach or Hilbert space. Given a vector "x" and a neighbourhood "U" of "x" we will study the return set "N_T(x,U) = { n : T^n(x) is in U }" and depending on its size, observed from the Combinatorial Number Theory point of view, we will say that the vector "x" presents one property of recurrence or another. The thesis memoir is a compendium of articles and it has four chapters and one appendix: 1. Adaptation of the revised "author version" of article "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), paper no. 109713, 36 pages". Here, the strong notions of reiterative, U-frequent and frequent recurrence are defined for the first time, and their basic properties are studied. The theory is finally generalized through the concept of F-recurrence, which is connected to the notion of F-hypercyclicity. 2. Adaptation of the revised "author version" of article "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". In this chapter the recurrence properties for linear operators are related to Ergodic Theory and measure preserving systems. 3. Adaptation of the revised "author version" of the preprint "Questions in linear recurrence: From the T+T-problem to lineability". We solve in the negative an open problem posed in 2014: Let "T : X ---> X" be a recurrent operator. Is it true that the operator "T+T" is recurrent on "X+X"? In order to do that we establish the analogous notion, for recurrence, to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and then we construct recurrent but not quasi-rigid operators on every separable infinite-dimensional Banach space. 4. Adaptation of the revised "author version" of the preprint "Recurrent subspaces in Banach spaces". In this chapter we study the spaceability (existence of an infinite-dimensional closed subspace) for the set of recurrent vectors. - Appendix. Looking for a self-contained text we have added an appendix with some of the basic Combinatorial Number Theory results that are taken for granted along the different chapters/articles forming this memoir. Following the regulations established by the Doctoral School the next sections are also included: - Introduction; - General discussion of the results; - Conclusions. / This thesis has been written at the “Institut Universitari de Matemàtica Pura i Aplicada” (IUMPA) of the “Universitat Politècnica de València” (UPV), during the period of enjoyment of a scholarship of the “Programa de Formación de Profesorado Universitario” granted by the “Ministerio de Ciencia, Innovación y Universidades”, reference number: FPU2019/04094. The research exposed has also been partially funded by the project “Dinámica de operadores” (MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00), thanks to which the author carried out a 3-month research stay in Lille, France (September-December 2021), that was supervised by Professor Sophie Grivaux; and also by the travel grant awarded by the “Fundació Ferran Sunyer i Balaguer” which allowed the author to carry out a 3-month research stay in Mons, Belgium (April-June 2023), supervised by Professor Karl Grosse-Erdmann. / López Martínez, A. (2023). Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/196101 / Compendio

This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla's and Pillai's theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman's inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam's problem and the Agoh-Giuga conjecture

Cette thèse est divisée en deux parties : la partie I traite de combinatoire additive, la partie II s’est portée sur des questions de théorie élémentaire des nombres. Dans le chapitre 1, on généralise la transformée de Davenport pour prouver que si S\mathbb A=(A, +)S est un demi-groupe cancellatif (éventuellement non commutatif) et SX, YS sont des sous-ensembles non vides de SAS tels que le sous semi groupe engendré par SYS est commutatif, on a SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, où S\gamma(\ctlot)S dénote la constante de Cauchy-Davenport d’un ensemble. On en obtient une extension des théorèmes de Chowla et Pillai pour les groupes cycliques et une version plus forte d’un théorème additif de Karolyi et Hamidoune. Dans le chapitre 2, on montre que si S(A,+)S est un semi-groupe cancellatif et si SX, Y\subsetcq AS alors SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. Cela donne une généralisation de l’inégalité de Kemperman pour les groupes sans torsion et une version plus forte du théorème d’Hamidoune-Karolyi. Dans le chapitre 3, on généralise des résultats par Freiman et al., en prouvant que si S(A,\ctlot)S est un semi-groupe linéairement ordonnable et SSS est un sous-ensemble fini de SAS engendrant un sous-semi-groupe non-abélien, alors S|S^2-\gc3|S|-2S. Dans le chapitre 4, on prouve des résultats liés à une conjecture par Gyorgy et Smyth sur la finitude des entiers Sn\gc1S tels que Sn^kS divise Sa^a \pmb^nS pour des entiers fixés SaS, SbS et SkS avec Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. Enfin, dans le chapitre 5, on considère une question de divisibilité dans les entiers, en quelque sorte liée au problème de Znam et à la conjecture d’Agoh-Giuga / This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla’s and Pillai’s theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman’s inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam’s problem and the Agoh-Giuga conjecture