More on Divisibility Criteria for Selected Primes

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].Naumowicz Adam - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandPiliszek Radosław - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics, 19(2):83-92, 2011. doi:10.2478/v10037-011-0014-5.C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012, 2000.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrz¸ebska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y.Adam Naumowicz. On the representation of natural numbers in positional numeral systems. Formalized Mathematics, 14(4):221-223, 2006. doi:10.2478/v10037-006-0025-9.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences

Using generalized binomial coefficients with respect to fundamental Lucas sequences we establish congruences that generalize the classical congruence of Wolstenholme and other related stronger congruences.Comment: 23 page

Computationally efficient search for large primes

To satisfy the speed of communication and to meet the demand for the continuously larger prime numbers, the primality testing and prime numbers generating algorithms require continuous advancement. To find the most efficient algorithm, a need for a survey of methods arises. Concurrently, an urge for the analysis of algorithms\u27 performances emanates. The critical criteria in the analysis of the prime numbers generation are the number of probes, number of generated primes, and an average time required in producing one prime. Hence, the purpose of this thesis is to indicate the best performing algorithm. The survey the methods, establishment of the comparison criteria, and comparison of approaches are the required steps to find the best performing algorithm. In the first step of this research paper the methods were surveyed and classified using the approach described in Menezes [66]. Wifle chapter 2 sorted, described, compared, and summarized primality testing methods, chapter 3 sorted, described, compared, and summarized prime numbers generating methods. In the next step applying a uniform technique, the computer programs were written to the selected algorithms. The programs were installed on the Unix operating system, running on the Sun 5.8 server to perform the computer experiments. The computer experiments\u27 results pertaining to the selected algorithms, provided required parameters to compare the algorithms\u27 performances. The results from the computer experiments were tabulated to compare the parameters and to indicate the best performing algorithm. Survey of methods indicated that the deterministic and randomized are the main approaches in prime numbers generation. Random number generation found application in the cryptographic keys generation. Contemporaneously, a need for deterministically generated provable primes emerged in the code encryption, decryption, and in the other cryptographic areas. The analysis of algorithms\u27 performances indicated that the prime nurnbers generated through the randomized techniques required smaller number of probes. This is due to the method that eliminates the non-primes in the initial step, that pre-tests randomly generated primes for possible divisibility factors. Analysis indicated that the smaller number of probes increases algorithm\u27s efficiency. Further analysis indicated that a ratio of randomly generated primes to the expected number of primes, generated in the specific interval is smaller than the deterministically generated primes. In this comparison the Miller-Rabin\u27s and the Gordon\u27s algorithms that randomly generate primes were compared versus the SFA and the Sequences Containing Primes. The name Sequences Containing Primes algorithm is abbreviated in this thesis as 6kseq. In the interval [99000,1000001 the Miller Rabin method generated 57 out of 87 expected primes, the SFA algorithm generated 83 out of 87 approximated primes. The expected number of primes was computed using the approximation n/ln(n) presented by Menezes [66]. The average consumed time of originating one prime in the [99000, 100000] interval recorded 0.056 [s] for Miller-Rabin test, 0.0001 [s] for SFA, and 0.0003 [s] for 6kseq. The Gordon\u27s algorithm in the interval [1,100000] required 100578 probes and generated 32 out of 8686 expected number of primes. Algorithm Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers invented by Doctor Verkhovsky [1081 verifies and generates primes and quasi primes using special mathematical constructs. This algorithm indicated best performance in the interval [1,1000] generating and verifying 3585 variances of provable primes or quasi primes. The Parametric Representation of Composite Twins algorithm consumed an average time per prime, or quasi prime of 0.0022315 [s]. The Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers algorithm implements very unique method of testing both primes and quasi-primes. Because of the uniqueness of the method that verifies both primes and quasi-primes, this algorithm cannot be compared with the other primality testing or prime numbers generating algorithms. The ((a!)^2)*((-1^b) Function In Generating Primes algorithm [105] developed by Doctor Verkhovsky was compared versus extended Fermat algorithm. In the range of [1,10001 the [105] algorithm exhausted an average 0.00001 [s] per prime, originated 167 primes, while the extended Fermat algorithm also produced 167 primes, but consumed an average 0.00599 [s] per prime. Thus, the computer experiments and comparison of methods proved that the SFA algorithm is deterministic, that originates provable primes. The survey of methods and analysis of selected approaches indicated that the SFA sieve algorithm that sequentially generates primes is computationally efficient, indicated better performance considering the computational speed, the simplicity of method, and the number of generated primes in the specified intervals

Preservice Elementary Teachers\u27 Understandings of Topics in Number Theory

Research suggests that preservice elementary teachers may lack the mathematics understanding necessary to teach mathematics for understanding. The literature has consistently linked student success in mathematics with teacher pedagogical content knowledge (PCK), and recent study suggested a link between teachers’ mathematical content knowledge and student achievement. There are gaps in the literature concerning preservice elementary teachers’ understanding of number theory, and little is known about how they develop number theory PCK or the relationship between their content knowledge and their PCK. The goals of this dissertation were to investigate the nature of mathematics concentration preservice elementary teachers’ content knowledge of number theory, the nature of their potential PCK in number theory, and the relationship between the two. To address these goals, I conducted a qualitative, interpretive case study of undergraduate students enrolled in a number theory course designed for preservice elementary teachers, using an emergent constructivist-based theoretical perspective. I gathered observational, interview, and document data and conducted analysis using constant comparative methods. Many of my findings concerning preservice elementary teachers’ understandings of number theory content pertain to their understandings of greatest common factor (GCF) and least common multiple (LCM). In particular, participants were more comfortable creating LCM story problems than creating GCF story problems, but their understandings of GCF story problems were closely related to the two meanings of division. In contrast to their understanding of story problems, participants were more comfortable with procedures for finding the GCF than with procedures for finding the LCM. In response to my other research questions, evidence suggests that preservice elementary teachers do possess potential PCK in number theory, namely potential knowledge of content and students and potential knowledge of content and teaching, and that they are related and influenced by specialized content knowledge, curricular content knowledge, experiences working with students, and epistemological perspectives. My data also suggest that preservice elementary teachers possess a type of PCK that is not explicitly represented by the literature, which I call general mathematical pedagogy. My findings hold many implications for practice. For example, data suggest a process through which preservice elementary teachers might develop a robust understanding of GCF story problems, which builds on their understandings of division. With this observed development process, instructors can scaffold preservice elementary teachers’ understanding of GCF story problems. My results also imply specific ways in which mathematics teacher educators and mathematicians may help preservice elementary teachers develop PCK in number theory. For example, instructors can pose hypothetical student conjectures and ask preservice elementary teachers to reflect on the knowledge necessary to teach the content, determine the validity of the conjecture, identify the concepts the student does and does not understand, suggest how they might respond to the student, and reflect on how they used their content knowledge to do so

Sets of numbers from complex networks perspective

Tesis por compendio[EN] The study of Complex Systems is one of the scientific fields that has had the highest productivity in recent decades and has not ceased to fascinate the community dedicated to studying its properties. In particular, Network Science has proven to be one of the most prolific areas within Complex Systems. In recent years, his methods have been applied to model multiple phenomena in real life, both naturally generated, such as in biology, and due to the actions and interactions of man, such as social networks or communication networks. Recently, it has been seen how the methods of Network Science can be applied in the context of mathematics, as is the case of Number Theory. One of the most studied cases is networks whose elements are numbers and which are related through the divisibility relation. The main objective of this thesis is to extend these studies to other sets of numbers. On the one hand, we study the divisibility in natural numbers when we obtain these from Pascal matrices of increasing size, which allows us to extract non-sequential sets of numbers with non-constant increments between them. On the other hand, we study the case of the divisibility relation of rational numbers. Cantor's diagonal argument provides a way to order all rational numbers, which allows us to check to what extent some of the properties observed for the divisibility of natural numbers are extensible to a more general context. The thesis is divided into 4 Chapters. Chapter 1 contains a general introduction to the thesis and it is structured into 6 sections. In Sections 1.1 and 1.2, we briefly introduce Network Science, show some application examples, and motivate the study of networks of numbers generated from the divisibility property. In Section 1.3, we define the objectives of this PhD thesis and its scope. In Section 1.4, we present the notion of network, its representations, and some measures that can be calculated on them, such as nodes degrees, their distribution, the assortativity and the clustering coefficients. In another hand, in Section 1.5, we review the best-known network models such as Erdo¿s and Re'nyi random networks, Watts and Strogatz small-world networks, Baraba'si and Albert scale-free networks, and hierarchical networks. Finally, at the end of this Chapter 1, we show in Section 1.6 a review of various studies carried out in order to apply Network Science methods to problems and properties that arise in Number Theory, such as divisibility networks or networks generated from Collatz's Conjecture. or Goldbach's Strong Conjecture. In Chapters 2 and 3, we show the results obtained and that have been published to date. Finally, in Chapter 4, we summarize the conclusions obtained and indicate some related problems that we consider of interest to address in the future.[ES] El estudio de los Sistemas Complejos es uno de los campos científicos que ha tenido mayor productividad en las últimas décadas y no ha dejado de fascinar a la comunidad que se dedica al estudio de sus propiedades. En particular, la Ciencia de Redes se ha mostrado como una de las áreas más prolíficas dentro de los Sistemas Complejos. En los últimos años, sus métodos han sido aplicados para modelar múltiples fenómenos de la vida real tanto generados de manera natural, como puede ser en el caso de la biología, como debidos a las acciones e interacciones del hombre, como puede ser el caso de las redes sociales o las redes de comunicaciones. Recientemente, se ha visto cómo los métodos de la Ciencia de Redes pueden ser aplicados en el contexto de las matemáticas, como es el caso de la Teoría de Números. Uno de los casos que más se han estudiado es el de las redes cuyos elementos son números y que se relacionan mediante la relación de la divisibilidad. El objetivo principal de esta tesis es extender estos estudios a otros conjuntos de números. Por una parte, estudiamos la divisibilidad en los números naturales cuando obtenemos estos a partir de subconjuntos de números naturales extraídos de matrices de Pascal de orden creciente, lo que nos permite extraer conjuntos de números de manera no secuencial y con incrementos no constantes entre ellos. Por otra parte, estudiamos el caso de la relación de divisibilidad de los números racionales, dado que a partir del argumento diagonal de Cantor se pueden ordenar, lo que nos permite comprobar hasta qué punto algunas de las propiedades observadas para la divisibilidad de los números naturales son extensibles a un contexto más general. La tesis se divide en 4 capítulos. El capítulo 1 contiene una introducción general a la tesis y está estructurado en 6 secciones. En las secciones 1.1 y 1.2, presentamos brevemente la Ciencia de Redes, mostrando algunos ejemplos de aplicación y motivamos el estudio de redes de números generadas a partir de la propiedad de divisibilidad. En la Section 1.3, definimos los objetivos de esta tesis doctoral y su alcance. En la sección 1.4, presentamos la noción de red, sus formas de representación y algunas medidas que se pueden calcular sobre ellas, como son los grados de los nodos, la distribución de estos grados, la asortatividad y los coeficientes de clustering. Por otro lado, en la Sección 1.5, revisamos los modelos de redes más conocidos como son las redes aleatorias de Erdös y Rényi, las redes de pequeño mundo de Watts y Strogatz, las redes libres de escala de Barabási y Albert y las redes jerárquicas. Mostramos en la Sección 1.6, una revisión de diversos estudios realizados con el fin de aplicar métodos de la Ciencia de Redes a problemas y propiedades que surgen en la Teoría de Números, como son las redes de divisibilidad o redes generadas a partir de la Conjetura de Collatz o la Conjetura Fuerte de Goldbach. En los Capítulos 2 y 3, mostramos los resultados obtenidos y que han sido publicados hasta la fecha y, finalmente, en el Capítulo 4, resumimos las conclusiones obtenidas e indicamos algunos problemas relacionados que consideramos de interés abordar en un futuro.[CAT] L'estudi dels Sistemes Complexos és un dels camps científiques que ha tingut major productivitat en les últimes dècades i no ha deixat de fascinar a la comunitat que es dedica a l'estudi de les seues propietats. En particular, la Ciència de Xarxes s'ha mostrat com una de les àrees més prolífica dins dels Sistemes Complexos. En els últims anys, els seus mètodes han sigut aplicats per a modelar múltiples fenòmens de la vida real tant generats de manera natural, com pot ser en el cas de la biologia, com deguts a les accions i interaccions de l'home, com pot ser el cas de les xarxes socials o les xarxes de comunicacions. Recentment, s'ha vist com els mètodes de la Ciència de Xarxes poden ser aplicats en el context de les matemàtiques, com és el cas de la Teoria de Números. Un dels casos que més s'han estudiat és el de les xarxes els elements de les quals són números i que es relacionen mitjançant la relació de la divisibilitat. L'objectiu principal d'aquesta tesi és estendre aquests estudis a altres conjunts de números. D'una banda, estudiem la divisibilitat en els nombres naturals quan obtenim aquests a partir de matrius de Pascal de grandària creixent, la qual cosa ens permet extraure conjunts de números de manera no sequëncial i amb increments no constants entre ells. D'altra banda, estudiem el cas de la relació de divisibilitat dels nombres racionals, atés que a partir de l'argument diagonal de Cantor es poden ordenar, la qual cosa ens permet comprovar fins a quin punt algunes de les propietats observades per a la divisibilitat dels nombres naturals són extensibles a un context més general. La tesi es troba dividida en 4 Capítols. El capítol 1, conté una introducció general a la tesi i está estructurat en 6 seccions. En les seccions 1.1 i 1.2, presentem breument la Ciència de Xarxes, mostrant alguns exemples d'aplicació i motivem l'estudi de xarxes de números generades a partir de la propietat de divisibilitat. En la Section 1.3, definim els objectius d'aquesta tesi doctoral y el seu abast. En la Secció 1.4, presentem la noció de xarxa, les seves formes de representació i algunes mesures que es poden calcular sobre elles, com són els graus dels nodes, la distribució d'aquests graus, la asortatividad i els coeficients de clustering. En la Sección 1.5, revisem els models de xarxes més coneguts com són les xarxes aleatòries de Erdös i Renyi, les xarxes de xicotet món de Watts i Strogatz, les xarxes lliures d'escala de Barabási i Albert i les xarxes jeràrquiques. Mostrem en la Sección 1.6 una revisió de diversos estudis realitzats amb la finalitat d'aplicar mètodes de la Ciència de Xarxes a problemes i propietats que sorgeixen en la Teoria de Números, com són les xarxes de divisibilitat o xarxes generades a partir de la Conjectura de Collatz o la Conjectura Forta de Goldbach. En els Capítols 2 i 3, vam mostrar els resultats obtinguts i que han sigut publicats fins hui i, finalment, en el Capítol 4, resumim les conclusions obtingudes i indiquem alguns problemes relacionats que considerem d'interés abordar en un futur.Solares Hernández, PA. (2021). Sets of numbers from complex networks perspective [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/176015TESISCompendi